Yes you can move.  Just make sure you forward your email to the new address.  Once you get your permit you don't need to switch it to the new county if you don't want to. I moved and didn't switch. I have a non restricted. If I switched the judge in the new county could put restrictions on it. I didn't want to risk that.

Question about pistol application, can you move to another county while your application is pending? Or would you have to start process over? Not planning on moving but some nice houses are popping up and would be nice to have the option to put an offer on one if it wouldn’t mess up my application