Big Bend National Park has NEVER taken phone reservations. You enter the park and go to the Panther Junction ranger station and reserve sites there. I'm talking what I call "car camping" sites: little cleared areas on dirt roads where you park and set up. Zero amenities. There are lots of them, they are secluded, and you can usually get the sites you want. Once there, you can reserve many days worth of sites, staying put, or moving around from site to site. Or, you can Gringo-out in the Chisos Mountains (a mountain right in the center of the park). Again, you have to go to the Chisos in person, but here you will rent a cabin, and they also have meals available. I've never done it, but I'm sure it's fine if a bit touristy. Wherever you stay, hiking up the Chisos to Emory peak is strenuous, but sort of a "must-do." Finally, there are two RV sites with hook-ups. Not sure how they work, but I am sure you have to book them in person.
Don't let the no pre-reservations thing dissuade you. Unless you go during spring break, there are always tons of available spaces. I once made the mistake of going during spring break, so I just got backpacker back country permits for the first couple of days. With one of them, you can park anywhere within the zone you claim that you will be backpacking in. However, with COVID, I don't think Spring Break is going to be an issue.
As for Big Bend Ranch State Park, they do take phone reservations, and I highly recommend the rustic luxury of staying a couple of nights at Saucedo, AKA the Big House. It's fairly popular, so make a reservation. The Ranch also has a "bunkhouse" where you can stay, and no-amenity car camping sites like in the National Park.
I also highly recommend the World Famous Litterbox Ranch.
