Do I have to buy a new iPhone 5s if I switch to AT&T?

My dad is considering switching from Verizon to AT&T but I wouldn't want to buy another 5S for $400 like the one I currently own.

The Verizon iPhone 5S actually should work on AT&T, since they contain universal technology and Verizon is required to unlock all 4G devices. That said, if you're signing up for a new post-paid contract, financially it might make more sense for you to sell your phone and buy a new one, since your new contract would be priced as if it were subsidizing the purchase of a new phone anyway.

No need to buy a new phone. If its locked to Verizon, just get them to unlock it (should be free of charge if out of contract). Sign up with AT&T then put the nanosim in.