Since parking isn't an issue until mid-May, might I suggest walking/hiking the National Seashore. Park at Coast Guard Beach in Eastham (not Orleans) and hike north to Nauset Light Beach. You can either hike back by beach or there is a road which is not very busy at this time of the year with some very nice views.
Alternatively, you can park at Nauset Light and reverse the trip to Coast Guard or hike north to Marconi Beach. There isn't a road back, so round-trip on the sand will be about 4.5 miles. Also, you can make the whole trek from Coast Guard to Marconi at about 6.5 miles r/t.
These three beaches are on the National Seashore (Atlantic side) below the cliffs.
When your done, stop by the Land Ho on Rte 6A & Main St. in Orleans for kale soup and a beer.