We actually do a bit of both each winter, we transit the US from Ontario to southern California and we take about a month stopping at state parks on the way there and back with a couple of months or so spent in one park in southern California. We've also moved from park to park all winter but the driving sometimes gets tiresome.
There are some great state parks with full/part services in Tennessee, Arkansas and Texas. Texas also has a park pass that saves $$ if you plan to stop in a few of their state parks.