Füssen is small enough that a car is not really needed. There is a bus to the castles and the trains to Munich and Augsburg and the old town is easy enough to access by foot. You can walk to Neuschwanstein and Hochschwangau castles via some marked roads; travel times posted on the signboards are comfortable over-estimates.
There are some great areas nearby, like some of the lesser-known castles, which would be easier to reach with your own car. It is possible to rent a car in Füssen.
Another alternative is to rent a mountain bike. There are numerous shops, one even right at the train station. Füssen has invested a lot into developing a network of bike routes and paths to make biking easier.
