We used to live in Houston and moved to Lubbock - I've been to the Palo Duro Canyon state park - it's a pretty impressive hole in the ground considering how freaking flat that whole area is - you can camp but all off-roading is limited to foot or bike. All the land around it is privately owned, there used to be a family that owned land with an entrance right near the state park entrance that did jeep tours in old wagoneers - some pretty scenic terrain. You used to be able to off-road around the Canadian river where it intersects 287 north of Amarillo. Otherwise there isn't much up there unless you want to head northwest up into New Mexico, but that's quite a way. Our route was through Killeen, brownwood, to lubbock - not the fastest but you get to see a little of the hill country.
Not much else up there unless you want to eat a 72 oz steak!
Watch out for deer at night this time of year.
Unless your dead set on Palo Duro, I'd head straight north to Gilmer in East TX and hit the wheeling at barnwell and then head up to super lift at hot spring Arkansas - prob shorter drive and much better wheeling and camping.