I was just in Big Bend and stayed at Cottonwood. It's a beautiful CG but my recollection is that Cottonwood is 100% NO reservations. So checking the reservation site may no do you any good.
In any case it did fill up by late afternoon (this is Jan). But if you arrive there in the morning you are pretty sure of finding a good spot.
I would spend one night just outside the park in Marathon then get up early and drive in to the park.
Cottonwood is great with HUGE sites right next to the river I enjoyed it a lot.