Cottonwood and RGV are very different but both are great, maybe a few nights in each? With that big Chevy, you should be able to drive most of the backroads but check conditions with staff first. The Old Ore Road leads to some awesome geologic features at Earnst Tinaja on the east end. The mercury mine ruins are also neat.
In the Basin, the Lost Mine and The Window are two of the signature trails in the Park. The South Rim can be done in a day if you're in good shape and start early but it's better as an overnight backpack. Emory Peak is a bit shorter and the highest point in the Big Bend.
On the west side, Santa Elena Canyon of course and sunset shots from Sotol Vista. The Old Maverick Road is usually passable and offers good photo ops in early morning and late afternoon.
Outside the Park, be sure to visit Terlingua. The Starlight Theater has decent food and La Kiva is a unique local watering hole worth a visit. The drive to Presidio is nice.