I've read that early Christians living in the Roman Empire changed the dates that holidays such as Easter and Christmas were celebrated to match the dates of Roman religious festivals to hide from Roman persecution. Is this true, and if it is why were the dates not moved back once Christianity became the official religion of the Empire?
I have never heard a credible argument that early Christians consciously changed the dates of their holidays to hide from persecution. The New Testament connects Easter to the Jewish holiday Passover, which is still more or less when it is.
It's much less clear historically why Christmas is when it is, since the NT doesn't say when exactly Jesus was born. It clearly is at the same time as Saturnalia and other pagan festivals, but it's much more likely that Christians had a tradition that Jesus was born around midwinter, and many people were both celebrating that and celebrating Saturnalia, and as more and more people became Christian, the pagan holidays were largely absorbed into Christmas.
I've written a few times about the Christmas issue. The main factors are:
i. little evidence that Christmas was meant to necessarily tie to the actual date of Jesus' birth as opposed to a date to commemorate that birth.
ii. the earliest clear dating for Christmas is the same as the feast of Sol Invictus, which means that the argument that Christmas was deliberately put at that time is no more convincing than vice versa.
iii. some difficult to evaluate evidence that the date of Christmas was established on the basis of Jesus' date of conception coinciding with Easter. I don't have all that argument to hand.
iv. the claim that Christians put Christmas on Dec 25th to co-opt a pagan holiday doesn't appear until very late (17th cent. iirc), which would be surprising if it were a deliberate strategy in antiquity.