The answer to both of your questions is NO. First, you don't become "saved" by being baptized. And secondly, since "Jesus" (although Christianity has twisted him into a Torah-less entity who came to abolish His Father's Torah) is actually the "same" Messiah, there's really no need to be re-baptized (unless you just want to!). Those who become Torah observant and realize there's more to God and the Bible, have simply "grown" from the watered-down Christian idea of "grace only" and learned to accept the fullness of who "Jesus" really is. People are certainly welcome to get "re-baptized" in the Name of Yeshua, if they wish, but it's not a requirement. We are, after all, speaking of YHWH, Yeshua and the Ruach haKodesh of the BIBLE - no matter what "versions" we are using.
Water baptism, as you said, is just an outward expression that one has accepted Yeshua as Messiah (which means they have also accepted YHWH as their God). The origin of baptism can be found in Leviticus 15:16 which shows that immersing in a mikveh (ritual) to become ritually clean. The following is from an appendix in the Aramaic English New Testament:
The majority of Christians are aware of the vital importance of a mikveh (baptism), which is a total immersion of the physical body in living water. This is an outward sign that is made after there is a change to the inner man. Once he has repented of his sin and dedicated himself to Mashiyach, there is no other action that is more indicative of one who has received the Good News than to be immersed into the Kingdom of Elohim. "And, my Brothers, I would have you know, that our fathers were all of them under the cloud, and they all passed through the sea; And they were all immersed by Moshe, in the cloud and in the sea; And they all ate the same spiritual food; And they all drank the same spiritual drink; for they drank from the spiritual rock that attended them, and that rock was the Mashiyach." (1 Corinthians 10:1-4).
All the Israelites who passed between the parted waters were immersed into the Good News. They had returned to their ancestral spiritual heritage of Awraham through the leading of Moshe Rabbeinu. "Your father Awraham did long to see my day, and he saw it and rejoiced!" (John 8:45 (56)). The Israelites were given every advantage to know and understand the Good News; their own deliverance after participating in the Passover lamb gave them a blood covering which was accepted before YHWH and a substitute for the life of their own first-born sons.
"And on the Shabbat day, we went without the gate of the city to the side of a river, because a house of prayer was seen there. And when we were seated, we conversed with the women who there assembled. And a certain woman who feared Elohim, a seller of purple, whose name was Lydia, from the city of Thyatira, (was there). Her heart our Master opened, and she listened to what Paul spoke. Then she was immersed, and her household..." (Acts 16:13-15).
The symbol of entering into a mikveh (living waters) and coming out purified was being practiced for centuries before Y'shua; the candidate arose from the waters renewed unto YHWH and born anew. Lydia "feared Elohim" and therefore presented herself to YHWH according to the requirement of the Word of YHWH (Torah). To Jewish onlookers Lydia's immersion at the side of the river was a mikveh based on Torah (the covenant of immersion [baptism] is Torah). "And some of them believed, and adhered to Paul and Silas; and of those Greeks who feared Elohim, a great many; and also of noted women, not a few." (Acts 17:4).
Water baptism is an act of obedience that follows conversion to portray symbolically the washing away of our sins. Water baptism calls for immersion instead of sprinkling. Many groups that have only sprinkled in times past are now seeing that the Scriptural method is total immersion in water. In fact, the Greek word "baptism" means to immerse, submerse and emerge.
The baptism itself does not bring salvation; salvation inspires the water baptism. It should be the first thing we do after we are "born again." Those who have become believers in Messiah Yeshua should desire to follow Him in this ordinance. Yeshua Himself came to John the Baptist to receive His water baptism because He was to take the sins of the world at His death. These sins would be washed away by YHWH for those who put their faith and trust in Him.
Yochanan (John) 1:29-31 - 29. And on the day that followed, Yochanan saw Y'shua who was coming towards him, and said, "Behold the Lamb of Elohim, He who takes away the sins of the world. 30. This is he whom I spoke concerning that, 'After me will come a man, yet he was before me because he is earlier than me.' 31. And I did not know him except that he be made known first to Yisrael. Because of this, I have come that I might immerse with water 32. And bear witness."
FOOTNOTE 1: his "Lamb" harkens back to the Akeida (binding of Isaac) when Isaac showed his willful obedience to lay down his life at the command of his father Avraham. Avraham and Isaac did not know then that YHWH would provide the ram caught in the thicket. Yitzak (Isaac) the son of Avraham and Sarah foreshadows Mashiyach ben Yoseph (the Son of YHWH via Yoseph and Miriam). YHWH provided Himself a "seh" lamb for the Perfect and complete sacrifice, not as the "ayil" (ram) caught in the thicket, but through the Lamb of YHWH, His only "begotten" Son Y'shua. The Akeida showed that YHWH provided Himself as the life that was in the place of Yitzhak's, as seen in the sin sacrifices in the temple system, "A life for a life."
There are many groups that view baptism as essential before someone is "saved" opposed to something that is done as an outward expression of faith after someone is saved. Many scriptures can go back and forth, but the one that we adhere to is from 1 Corinthians where Paul says:
1 Cor 1:14-17 - 14. I thank my Elohim that I immersed none of you, except Crispus and Gaius; 15. And therefore that not any one should say that I immersed in my own name. 16. I moreover immersed the household of Stephanas: but further, I know not that I immersed any other. 17. For Mashiyach did not send me to immerse, but to preach; not with wisdom of words, or else the stake of Mashiyach should be inefficient. 18. For the word concerning the stake is, to them who perish, foolishness; but to us who live, it is the energy of Elohim. 19. For it is written: I will destroy the wisdom of the wise; and I will dissipate the intelligence of the sagacious.
Notice that Paul, in verse 17, does not place "baptize" with the "gospel" that Messiah sent him to preach. If baptism was essential for salvation, then surely that would have been a part of the gospel and Paul would have taken great pleasure in baptizing everyone he shared the truth of Yeshua with.
As far as John 3:5 is concerned, there are several ways that this verse can be interpreted based on one's belief about baptism, the Holy Spirit, salvation, etc. It is next to impossible to use this verse as a foundation verse. The foundation of belief needs to be built on other verses.
However, the symbol of entering into a mikveh (living waters) and coming out purified was being practiced for centuries before Y'shua; the candidate arose from the waters renewed unto YHWH and born anew. Lydia "feared Elohim" and therefore presented herself to YHWH according to the requirement of the Word of YHWH (Torah). To Jewish onlookers Lydia's immersion at the side of the river was a mikveh based on Torah (the covenant of immersion [baptism] is Torah). "And some of them believed, and adhered to Paul and Silas; and of those Greeks who feared Elohim, a great many; and also of noted women, not a few." (Acts 17:4).