Rio Grande Village Campground has near 100 dry camping spots. Most will hold your near 35 foot 5th wheel just fine.
Rio Grande Village Store concessionaire operates the parking lot FHU 'campground'. It's nothing fancy, but it is relatively cheap for a NPS FHU campground.
Cottonwood CG will hold your 5er, but it is a 'no generators allowed' campground.
Chisos Basin CG does not have any spots that will hold your 5er. (Yes my sig pic was taken in that CG, but I was one of the hosts in the FHU site.)
Be aware there are no showers in the restrooms in Big Bend NP. The Rio Grande Village Store does operate a small pay shower facility, and a small coin laundry.
Also be away that Big Bend's busy season is Nov-Apr - with the park very crowded in March with spring break visitors.
Off the park proerty are two popular campgrounds for visitors near Study Butte/ Terlingua - the campground behind the single gas station. BJ's RV park about three miles toward Lajitas is also used by many folks. Both have FHU. About 20 miles farther south in Lajitas is Maverick Ranch - another bare lot campground but with some amenities. Many larger rigs like Maverick very much.
Be aware that one powerline comes into the area, and it an issue develops the entire area is without power. We once went 70 hours with no power in the park, Study Butte/ Terlingua/ Lajitas when there was a big problem near Alpine.
There is a nice small store just outside the park in Study Butte - Cottonwoods. There a typical in-park items at the stores at Rio Grande Village, Chisos Basin, Castolon, and a few items at the gas station near the park HQ at Panther Junction. There is fuel at RGV and Panther Junction. Expect to pay about $1 per gallon more than Fort Stockton. There is fuel in Study Butte and Lajitas - about 60-75 cents per gallon more than Fort Stockton.
Otherwise, it is 120-160 miles to Apline or Presidio for any supplies you didn't bring with you. It is also 120-160 miles from any place in the park to Alpine for a hospital/ doctor. There are two pharmacies in Alpine, but both are small town independents - they are only open til 5-6 pm weekdays, half-day Saturday and closed on Sunday. (The nearest 24-hour pharmacy is 280 miles away in Odessa.)
Another dry camping option are one of
Primitive Roadside Camping SitesLook at the pictures of each site / location and a map for the roads into the area. All are desert camping with low scrub around the cleared camping area. A different experience more like west coast/ southwest states boondocking than typical Texas 'remote' camping.
I would take my 36' TT to Grapevine Hills 1, 2 or 3. There is no way the rig could get to GH 4 or 5 due to a steep wash that has to be crossed.
I could go into Rattlesnake Mountain on the Old Maverick Road from the north, but I wouldn't take the trailer south of Luna's Jacal. And the Ocotillo Grove and Terlinqua Abajo sites area is almost too small for my truck alone.
As with any backcountry campsite on barely improved roads - make a recon drive without the trailer first.
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