Some quick answers.
Why did John the Baptist continue his separate ministry and group of disciples if he recognized Jesus as God incarnate?
John 3:
29....Therefore this joy of mine is
fulfilled.
30 He must increase, but I
must decrease.
In the case where he didn't fully recognize Jesus as the Messiah, why did he have to send messengers to determine this? Was he Jesus's cousin?
John's mother Elizabeth and Mary were cousins. In the reading of all the Gospels,
one thing that becomes quite evident, is that many who were waiting for Messiah
had particular expectations.
In this case, they thought the Messiah would restore the kingdom of Israel...NOW.
Instead, John found himself in prison. When he sent disciples to inquire of Christ,
what he was saying is this, "aren't you the promised one, the Messiah? If so, why
am I in prison? We cannot allow personal expectations to dictate would God alone controls.
Why did his disciples exist as a separate competing sect after Jesus's crucifixion?
Every effort was made to avoid conflict.
John 3:
22 After these things Jesus and His disciples came into the land of Judea, and there
He remained with them and baptized.
23 Now John also was baptizing in Aenon near Salim, because there was much water there.
And they came and were baptized.
24 For John had not yet been thrown into prison.
25 Then there arose
a dispute between some of John’s disciples and the Jews
about purification.
n
ote: Many were still seeking John to repent and be baptized. Apparently this was acceptable
and unopposed by Christ. The dispute was between his disciples and other Jews, not the
disciples of Jesus. They used the opportunity to question why Jesus and his disciples were
also baptizing. Why Not? Jesus did not baptize anyway, he left that to His disciples.
John 4:
1 Therefore, when the Lord knew that the Pharisees had heard that Jesus made
and baptized more disciples than John.
2 (though Jesus Himself did not baptize, but His disciples),
3 He left Judea and departed again to Galilee.
Back to John 3:
26 And they came to John and said to him, Rabbi, He who was with you beyond the Jordan,
to whom you have testified—behold, He is baptizing, and all are coming to Him!
notice they refer to Jesus as "He who was with you".
LOL, I guess that had that reversed.
27 John answered and said, A man can receive nothing unless it has been given to him from heaven.
28 You yourselves bear me witness, that I said, ‘I am not the Christ,’ but, ‘I have been sent before Him.’
29 He who has the bride is the bridegroom; but the friend of the bridegroom, who
stands and hears him, rejoices greatly because of the bridegroom’s voice.
Therefore, this joy of mine is fulfilled.
It is hard to imagine John and His disciples not becoming followers of Jesus.
John must have realized that he had fulfilled his calling after baptizing
Jesus "to fulfill all righteousness".
There were a few who remained loyal to John, whether or not that was
misguided is only to surmise. After John's death, many may well have joined
with Jesus and His disciples, but that also is speculating,