"And he commanded them to be baptized in the name of the Lord. Then they asked him to stay a few days." (Acts 10:48)
"When they heard this, they were baptized in the name of the Lord Jesus." (Acts 19:5)
There are many passages in the New Testament
about doing things "IN THE NAME of the Lord" or "IN THE NAME of Jesus" (casting out demons and teaching "IN THE NAME
of Jesus") and no where else is there Biblical instruction to do anything "IN THE NAME
of the Father" or "IN THE NAME of the Holy Spirit". For instance:
"And whatever you do in word or deed, do all in the name of the Lord Jesus, giving thanks to God the Father through Him." (Colossians 3:17)
If you do a word search, using Strong's Concordance or a computer program, using the phrase "in the name of" you'll find no reference to actions taken "in the name of the Father" or "the name of the Holy Spirit".
I also checked the original Greek scripture for Matthew 28:19 and there is no other way to translate the actual Greek words of that passage. Nor are there any footnotes in any of the translations I can find designating the passage as "suspicious" or "questionable". See Mark 16:9-20 and 1John 5:7-8 which some translators say were either not there in the oldest manuscripts (the former) or slightly different in original manuscripts (the latter).
Another fact to consider is that all our English language Bibles are translations from the Greek and Hebrew (or Aramaic) languages and that syntax (the way words are put together to form phrases, clauses or sentences) is completely different in English, Greek and Hebrew (and many other languages). So there may be punctuation, or phrasing differences in the syntax which would change the way this passage reads:
"Go therefore and make disciples of all the nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit, " (Matthew 28:19)
However, I have no Biblical or secular justification for saying that the above passage contains any error. Should we use this single passage to justify baptizing a believer (or ourselves being baptized) using that phrase? Each of us will have to make that decision on their own. God is understanding and forgiving. Certainly we would not be blaspheming to use the phrase.
The baptism of a believer should be done "in the name of Jesus", corresponding to Colossians 3:17 and all the other passages which use this term, including this one:
"Then Peter, filled with the Holy Spirit, said to them, "Rulers of the people and elders of Israel: If we this day are judged for a good deed done to a helpless man, by what means he has been made well, let it be known to you all, and to all the people of Israel, that by the name of Jesus Christ of Nazareth, whom you crucified, whom God raised from the dead, by Him this man stands here before you whole. This is the 'stone which was rejected by you builders, which has become the chief cornerstone.' Nor is there salvation in any other, for there is no other name under heaven given among men by which we must be saved." " (Acts 4:8-12)
In summary, it is never a good idea to base an idea or a doctrine on a single passage of scripture. The Bible is "multiple - redundant", i.e., all the things that are vital to our salvation and our knowledge of God are repeated throughout the Old and the New Testament canon so often that if one studies and compares scriptures, we may find that a single passage is misunderstood, or mistranslated or that it is not important to our salvation. So it is with this passage.
The important thing is to believe on Jesus, repent of our sins, confess His name before men and to be baptized for the remission of our sins and to come up from that symbolic grave with a determination to change our sinful ways, to learn the ways of God and to live by His commandments to the best of our ability, with prayers to Him to help us live that way and for greater understanding of what He would have us do.