Why don't you take the train from Rome to Salerno ? There are direct fast trains until 19.45 (after that only regular trains) and it takes 2.5 hours. Then you can stay in Salerno 1 night and rent a car in Salerno the day after.

Driving from Rome to Salerno there is the Highway (autostrada) but I'm wondering about when you get off the autostrada to look for small towns during the night. Moreover you are so tired.

Anyway if you spend the night driving you will be so tired that you lose all the following day sleeping so it could be better to take the train to Salerno, arrive at a reasonable time in the city and rest in a hotel. The following morning you will be enough rested to look for a convenient seaside town.

Ciao, Marco in Milan - Italy