The word, "believe," appears 119 times in the Gospels. The word, "baptize," appears only about half as many times, and of those, 41 speak of the baptizing work of John, 12 speak of the "baptism of the cross," and 8 refer to the Lord's disciples baptizing. All those baptized in the Gospels were baptized unto repentance, not into the Church, as it did not begin until Acts 2.
Mark 16:16 does not say a person who believes, and yet has not been baptized is damned; it says he is damned if he does not believe.
John 3:5 is referring to the physical birth (as in "my water broke"), in contrast with the baptizing work of the Holy Spirit, whereby a believer is baptized into Christ (Rom. 6:3; 1 Cor. 12:13; Gal. 3:27; Eph. 4:5). In other words, be "born again."
Acts 2:38 says that those Jews who believed the Gospel, should repent and be baptized for (because of, or due to) the remission (payment already accomplished by Jesus on the cross).
Acts 22:16 says ". . . arise, and be baptized, and wash away thy sins, calling on the name of the Lord." Ananias, most likely a Jewish proselyte (v. 12-14), was told to ceremonially wash, as did those of John's baptism, including Jesus (who had no sins to wash away - Heb. 4:15). Notice Paul did not tell us that we were to be baptized in Romans 10:8-13 in order to "wash away our sins," or to be saved.
1 Peter 3:20 speaks of Noah's flood being a type of baptism, speaking of Noah being saved "by" water. The Greek word is "διά" (dia), which could, and perhaps should be translated "from," as it is in 2 Thessalonians 2:2. 1 Peter 3:21 says, "The like figure whereunto even baptism doth also now save us (not the putting away of the filth of the flesh, but the answer of a good conscience toward God), by the resurrection of Jesus Christ". Notice that it is not a washing away of sins, but a cleansing of one's conscience. It does not save, but produces inner peace.
Baptism does not save; saved people are baptized!