Geesh, Sorry you have to go through it again. Sounds like you're a little paranoid because of the first time around. I would be too. But it does sound like this mom and dad are better so it should go better.
I would think you are going to HAVE to move them. And it is just a myth about touching babies and the parents not wanting anything to do with them again.Just be very gentle when you do it.
Hopefully someone with experience will be on to give you some advice. Or you could always call your Avian vet and they should be able to advise you