Must I Start RCIA Over?

Not necessarily. Your current pastor could possibly let you finish via viewing recorded lectures online or by letting you finish via directed distance learning. That way, you could move to your new parish but return to your old parish to be received into the Catholic Church—or into full communion with the Church if you’re already baptized.

As an alternative, your pastor could verify with your new pastor that you have already completed a significant part of RCIA so that your new pastor could make accommodations for you. One way or other, I’m hopeful that the two pastors could work something out that would preclude your having to start RCIA all over again.