I'm heading out to CA from KY this summer, and I typically camp nights to save some cash. After an unfortunate and inconvenient midnight awakening in the (apparently privately owned) backwoods of North Carolina, I am being more proactive about camping exclusively where it is allowed.
I'm looking for a free spot to pitch a tent near Amarillo, Texas. Front yards with permission, city parks where it is allowed, it all works. My one lead is that possibly the "Amarillo Welcome Center" has free tent sites available? Anywhere within 100 miles farther east of Amarillo along I 40 would work well too.
Check out lake Meredith national recreation area. I think you can camp there and it's about 40 miles from Amarillo. It may be free, but if its not it'll likely be cheap.
Palo duro canyon state park is beautiful, but you'll have to pay a small fee to camp there. It is worth the cost to visit there though. It's only about 20 miles from Amarillo.
It isn't free... but we're I in your shoes... I'd look south from Elk City, OK down to Granite to camp and climb Quartz Mountain... Top grade slab climbing, great boulders. The Welcome center might be your only free option. I'll ask if any of the locals know more and get back if they have anything to add. Good luck.
The state park has a pull over before the main gate. It is reserved for after hour travelers. They are okay with people pitching a tent next to their car if your arrive late at night. Hope this helps.