If the new P-trap is plastic, you don't need putty, and absolutely don't use pvc cement unless you want to replace it every time you need to clean the trap!
The seals should be fine. The only place you should be concerned is where it transitions into the main waste. The transition fitting should be cemented to the waste pipe, with the proper seals there. If its a metal nut to a metal transition, make sure its clean and the seal is positive on that pipe (that's the only place I've ever used putty - transitioning a new pvc p trap to an ancient pipe on a house built in the 1930's)