You didn't give the month of the year. This area is very crowded from May to October, and you will need to book well in advance. Public transportation can be crowded as well, particularly with the curby, single, two-lane road that goes from Sorrento to Salerno, more or less. Busses have to stop traffic to make it around curves on this road. That's why you don't want to drive it.
Many people, ourselves included, stay in Sorrento because it's the LAST stop on the local commuter train, giving good access to Naples, and the five ancient excavation sites.
You should look up some old posts here to learn just how far (in hours) it is to Paestum, and how annoying it is to drive and park along the Amalfi Coast. Even in Sorrento (not on the AC) we saw long lines of cars outside underground garages every night that we walked to dinner. We wanted to go to Paestum, but it is too hard by public transportation. You are going to use that as justification for keeping your car, but I don't enjoy sitting in a long line of traffic for an hour or more.
I don't have teens, but I wonder if a full day to see Paestum will be interesting for them? Sorry to sound cynical, I just didn't know what an important place "was" when I was a teenager.
Do you understand that many hotels are up or down dozens of steps in the most popular AC towns? I would get parking promises very clear before selecting a hotel on that basis.