Seems like you are getting a lot of people telling you not to worry that it won't be busy in the fall. We typically camp in Moab 3 to 4 weeks every year, probably far more than most on this forum. It is just over three hours from our house so it's a quick get away for us. Fall has gotten to be a popular time to go camping there, especially before the ski resorts open in mid-November. (We've camped in Moab during Thanksgiving week several times when the weather was quite pleasant and the skiing not quite ready).
But let me reiterate that the weekend of the Utah Teachers Convention (October 15-19 this year) is not an easy weekend to find a BLM campsite in Moab. For instance, one year we arrived out there around noon to find every BLM and NPS campsite full. We spent hours driving to all the places I knew of. We finely ended up high in the LaSal Mountains at Warner Lake CG. Luckily for us, it hadn't snowed up there yet as it is inaccessible once it does.
Whenever our local school has a long weekend in October that coincides with the Utah Teachers Convention weekend, we now don't go to Moab. We've been to Vernal several times instead.
Now others may have had a different experience on that weekend, but I'd still strongly recommend arriving early Wednesday morning or even better on Tuesday if you want to find a spot.