The basic answer is yes for full immersion baptism but not necessarily always "by the hand of others." However, this does not mean that sprinkling wasn't also used. Evidence for immersion baptism is found, even before the advent of Christianity, in the Jewish tradition for Gentile converts. This tradition was probably inherited by the earliest (pre-Pauline) churches, since they saw themselves not as a new religion, but as Jews who had found the Messiah.
The Jewish Ecyclopedia informs us that Jewish baptism was understood
as a religious ablution signifying purification or consecration... The
Rabbis connect with this the duty of bathing by complete immersion
("ṭebilah," Yeb. 46b; Mek., Baḥodesh, iii.).. these three acts
[washing one's clothes, sprinkling with blood and full immersion in
flowing water] being the initiatory rites always performed upon
proselytes, "to bring them under the wings of the Shekinah" (Yeb.
The tradition of water baptism in a flowing stream was also used by John the Baptist, but administered to Jews, not converts to Judaism. The disciples of Jesus, several of whom had followed John previously, certainly inherited the tradition from him.
The J.E. article continues:
Baptism, next to circumcision and sacrifice, was an absolutely
necessary condition to be fulfilled by a proselyte to Judaism (Yeb.
46b, 47b; Ker. 9a; 'Ab. Zarah 57a; Shab. 135a; Yer. Kid. iii. 14,
64d)... The next ceremony, adopted shortly after the others, was the
imposition of hands, which, it is known, was the usage of the Jews at
the ordination of a rabbi.
The laying on of hands was another ceremony adopted by those Jews who followed Jesus and later inherited by Christianity. But regarding baptism itself being administered only by the hand of another, as opposed to by oneself, this is less certain. Jewish ritual baths are entered by oneself, but with a witness, whose function is to declare the rite properly accomplished. On the other hand, John 4 reports
Now when the Lord [Jesus] knew that the Pharisees had heard that Jesus
was making and baptizing more disciples than John (although Jesus
himself did not baptize, but only his disciples), he left Judea and
departed again to Galilee.
Since the text specifies that Jesus himself did not participate, we may understand that the disciples did take an active role in baptizing new converts. In other words, they assisted converts manually. The most likely method would be to use the same as John's, immersion in flowing water. However, since in the Jewish rite the laying on of hands is a separate rite, as it was in early church, there does not seem to be any particular significance to touching as part of baptism. It is the water, not than hands, that is important.
Although being baptized by another [either standing by or manually assisting] was probably the norm, we do know of at least one Christian saint, Thecla, who baptized herself. Facing martyrdom with having been baptized:
...when she had ended her prayer, she turned and saw a great tank full of
water, and said: Now is it time that I should wash myself. And she
cast herself in, saying: 'In the name of Jesus Christ do I baptize
myself on the last day.'
Thecla's case was accepted by the church as a special in extremis situation but self-baptism was definitely not the norm, both because it was by this time frowned upon, and because she was a woman; and therefore not qualified to administer the rite.
It should be noted that although the earlier examples provide evidence for full-immersion baptism "in the time of Jesus in Judea" this does not mean that sprinkling was not practiced as well. Sprinkling and hand-washing were part of the Levitical purity rites, and the prophet Ezekiel had declared: "Then will I sprinkle clean water upon you, and ye shall be clean." (36:25)
From the above we know that full-immersion baptism was practiced by converts to Judaism in Judea, and this is the likely source of Christian baptism as well. Converts were manually assisted by a minister in Christian baptism, but this was probably not a necessary part of the rite.