Im not speaking for Dave but ive traveled many of the back roads in Big Bend NP and typically the back country roads are in pretty good shape. Obviously, check at the entrance for current conditions
The River Road is well maintained with many great campsites and there are a number of roads off there that are ready to travel. Black Gap is pretty technical but not bad if you're careful.
I'll be there next week and can post conditions.
Unless it rains or has rained or is going to rain on you while out on the dirt roads, then River Road becomes "sporty" in some spots, Old Ore and Mavick aren't too bad.
For backcountry sites I like Terlingua Abajo or Pine Canyon (especially #5, see attached photo from that campsite) a lot more than the River Road sites. Grapvine Hills, by the way, seems perfect, except that dirt road gets a LOT of traffic all day and past sunset due to the popular trail down the road a bit.
Only once have I had a dangerous encounter with someone in Big Bend who crossed the Rio Grande and that was in Boquillas Canyon along the trail. The "upside" was that the Ranger at the Rio Grande Village Station (first one we found) only came out to chat about the incident because I was a peace officer at the time and stated I was going to go file a report with the SO. He simply didn't care because it was so common at the time (2010-ish). The last few times we've visited (five times in the past 4 years) we've had no issues with anyone except Jeep Bro's acting like Jeep Bro's out on the dirt roads and such.
I take my 2wd van on all the roads in Big Bend, just be smart, have a friend, have a dual-band radio (the linked repeater network in the region is amazing) and even with the best antenna and a Wilson repeater, there are sections where no cell service is available...period.
If you want a not-backcountry site, make reservations now (and it might be too late). For winter camping I highly recommend Cottonwood Campground. Quieter than The Basin, warmer (usually) and the birding is amazing. If not, then hit The Basin for epic star fields and sunsets while in camp. If you can't score a camp in BBNP or the State Park, check out Black Gap, it is a lot wilder than the two parks and ground fires are allowed as well (weather depending). There are numerous private campgrounds in and around Terlingua, which will make it easier to have a beverage on the porch of the Starlite for sunset over the ghost town and cemetary, which I also recommend.
Standing about 20ft in front of the van at our Pine Canyon #5 site last December.
About 10ft in front of my tent in The Basin campground July a few years ago (you can see the outline of Casa Grande).
South to Big Bend I'd then checkout Ft. Davis area and parks. Further south to Big Bend! Don't forget Big Bend area has the State Park too. Not as touristy but better camping and trails you can also have fires at campsite. I'm in Central TX near Waco but shout if you have any other questions. Just remember Texas is a "Whole other country" a lot to do but VERY BIG 
