Numbers 9:10-13 was documented because there were a number of people who had come to Moshe because they were unclean because of a corpse:
6 But there were certain people who had become unclean because of someone's corpse, so that they could not observe Pesach on that day. So they came before Moshe and Aharon that day 7 and said to him, "We are unclean because of someone's corpse; but why must we be kept from bringing the offering for ADONAI at the time designated for the people of Isra'el?" 8 Moshe answered them, "Wait, so that I can hear what ADONAI will order concerning you." 9 ADONAI said to Moshe, 10 "Tell the people of Isra'el, 'If any of you now or in future generations is unclean because of a corpse, or if he is on a trip abroad, nevertheless he is to observe Pesach. 11 But he will observe it in the second month on the fourteenth day at dusk. They are to eat it with matzah and maror, 12 they are to leave none of it until morning, and they are not to break any of its bones -they are to observe it according to all the regulations of Pesach.
Notice the last verse: they are to observe it according to all the regulations of Pesach.
The Pesach was originally described in Exodus 12:3-20, and includes the regulations of Pesach:
3 Speak to all the assembly of Isra'el and say, 'On the tenth day of this month, each man is to take a lamb or kid for his family, one per household - 4 except that if the household is too small for a whole lamb or kid, then he and his next-door neighbor should share one, dividing it in proportion to the number of people eating it. 5 Your animal must be without defect, a male in its first year, and you may choose it from either the sheep or the goats. 6 "'You are to keep it until the fourteenth day of the month, and then the entire assembly of the community of Isra'el will slaughter it at dusk.
7 They are to take some of the blood and smear it on the two sides and top of the door-frame at the entrance of the house in which they eat it. 8 That night, they are to eat the meat, roasted in the fire; they are to eat it with matzah and maror. 9 Don't eat it raw or boiled, but roasted in the fire, with its head, the lower parts of its legs and its inner organs. 10 Let nothing of it remain till morning; if any of it does remain, burn it up completely. 11 "'Here is how you are to eat it: with your belt fastened, your shoes on your feet and your staff in your hand; and you are to eat it hurriedly. It is ADONAI's Pesach [Passover]. 12 For that night, I will pass through the land of Egypt and kill all the firstborn in the land of Egypt, both men and animals; and I will execute judgment against all the gods of Egypt; I am ADONAI.
13 The blood will serve you as a sign marking the houses where you are; when I see the blood, I will pass over [Hebrew: pasach] you - when I strike the land of Egypt, the death blow will not strike you. 14 "'This will be a day for you to remember and celebrate as a festival to ADONAI; from generation to generation you are to celebrate it by a perpetual regulation. 15 "'For seven days you are to eat matzah - on the first day remove the leaven from your houses. For whoever eats hametz [leavened bread] from the first to the seventh day is to be cut off from Isra'el. 16 On the first and seventh days, you are to have an assembly set aside for God. On these days no work is to be done, except what each must do to prepare his food; you may do only that.
17 You are to observe the festival of matzah, for on this very day I brought your divisions out of the land of Egypt. Therefore, you are to observe this day from generation to generation by a perpetual regulation. 18 From the evening of the fourteenth day of the first month until the evening of the twenty-first day, you are to eat matzah. 19 During those seven days, no leaven is to be found in your houses. Whoever eats food with hametz in it is to be cut off from the community of Isra'el - it doesn't matter whether he is a foreigner or a citizen of the land. 20 Eat nothing with hametz in it. Wherever you live, eat matzah.'"
So in verses 14 and 19, we see that the regulations for the 7 days of unleavened bread are part of the observance of Pesach. So, yes, if you are on a journey and can't observe Pesach on its proper date, you observe it on the 14th of the 2nd month, as well as rid your abode of leaven, and eat only matzah for a week. But the counting of the omer and the observance of Shavuot is NOT changed - you still do that despite your missing Pesach.
This rationale is confirmed in 2 Chronicles 30:
1 Then Hizkiyahu sent to all Isra'el and Y'hudah, and wrote letters also to Efrayim and M'nasheh, summoning them to the house of ADONAI in Yerushalayim, to keep the Pesach to ADONAI the God of Isra'el. 2 For the king, his officials and the entire Yerushalayim community had agreed to keep the Pesach in the second month. 3 They had not been able to observe it at the proper time because the cohanim had not consecrated themselves in sufficient number; also the people had not assembled in Yerushalayim. 4 The idea had seemed right to the king and to the whole community; 5 so they issued a decree that it should be proclaimed throughout all Isra'el, from Be'er-Sheva to Dan, that they should come to keep the Pesach to ADONAI the God of Isra'el at Yerushalayim; for only a few had been observing it as prescribed.
6 So runners went with the letters from the king and his officers throughout all Isra'el and Y'hudah. They conveyed the king's order: "People of Isra'el! Turn back to ADONAI, the God of Avraham, Yitz'chak and Ya'akov! Then he will return to those of you who remain, who escaped capture by the kings of Ashur. 7 Don't be like your ancestors, or like your kinsmen who sinned against ADONAI the God of their ancestors, with the result that he allowed them to become an object of horror, as you see. 8 Don't be stiffnecked now, as your ancestors were. Instead, yield yourselves to ADONAI; enter his sanctuary, which he has made holy forever; and serve ADONAI your God; so that his fierce anger will turn away from you. 9 For if you turn back to ADONAI, your kinsmen and children will find that those who took them captive will have compassion on them, and they will come back to this land. ADONAI your God is compassionate and merciful; he will not turn his face away from you if you return to him."
10 So the runners passed from city to city through the territory of Efrayim and M'nasheh, as far as Z'vulun; but the people laughed at them and made fun of them. 11 Nevertheless, some from Asher, M'nasheh and Z'vulun were humble enough to come to Yerushalayim. 12 Also in Y'hudah the hand of God was at work, uniting their hearts to do what the king and the leaders had ordered in accordance with the word of ADONAI. 13 Thus, many people assembled in Yerushalayim to keep the festival of Matzot in the second month, a huge crowd. 14 First they set about removing the altars that were in Yerushalayim, and they also removed all the altars for incense and threw them in Vadi Kidron. 15 Then they slaughtered the Pesach lamb on the fourteenth day of the second month. Ashamed of themselves, the cohanim and L'vi'im had consecrated themselves and brought burnt offerings into the house of ADONAI. 16 Now they stood at their stations, as prescribed in the Torah of Moshe the man of God; the cohanim splashed the blood given to them by the L'vi'im.
17 For there were many in the assembly who had not consecrated themselves; therefore the L'vi'im were responsible for slaughtering the Pesach lambs and consecrating them to ADONAI on behalf of everyone who was not clean. 18 For a large number of the people, especially from Efrayim, M'nasheh, Yissakhar and Z'vulun, had not cleansed themselves but ate the Pesach lamb anyway, despite what is written. For Hizkiyahu had prayed for them, "May ADONAI, who is good, pardon 19 everyone who sets his heart on seeking God, ADONAI, the God of his ancestors, even if he hasn't undergone the purification prescribed in connection with holy things." 20 ADONAI heard Hizkiyahu and healed the people. 21 The people of Isra'el there in Yerushalayim observed the festival of Matzot for seven days with great joy; while every day the L'vi'im and cohanim praised ADONAI, singing to ADONAI with the accompaniment of loud instruments.
Never mind these circumstances, that is, whether or not the people had touched a corpse or were considered on a journey. In this case, this was a "mass return" to YHWH, for many had fallen away - see verses 6-12 immediately above. In 2 Chronicles 29, we find that King Hizkiyahu "2...did what was right from ADONAI's perspective, following the example of everything David his ancestor had done. 3 In the first month of the first year of his reign, he reopened the doors of the house of ADONAI and repaired them."
And: "16 The cohanim went in to cleanse the inner part of the house of ADONAI; all the unclean things they found in the sanctuary of ADONAI they brought out into the courtyard of the house of ADONAI, where the L'vi'im took and carried them out to Vadi Kidron. 17 They began consecrating on the first day of the first month, and on the eighth day of the month they reached the vestibule of ADONAI. Then they consecrated the house of ADONAI in eight more days; so that on the sixteenth day of the first month, they had finished."
So you see, that though King Hizkiyahu was working hard to restore righteousness, and to get the temple ready, they missed the date of Pesach in the 1st month. That brings us to 2 Chronicles 30, verse 21, where the people also celebrated the 7 days of matzah when they observed Pesach in the 2nd month, confirming that the Feast of Unleavened Bread is part of the observance of Pesach.