The winter is the busy time which you have indicated you can tell from your checks of the Recreation.gov website. However the really full time is spring break, and you will be before that.
You should not have an issue getting a campsite at Cottonwood. Check at the entrance station or the Park HQ at Panther Junction when you arrive. They will know in the CG is full or has been full in recent days. Chisos Basin is the CG which fills first most of the time.
Be aware there are no showers at any NPS campground in the park and water is in short supply. I would advise arriving with enough bottled water for drinking and cooking needs for your stay.
While bear have been sighted a couple times near Santa Elena Canyon in recent years, they do not frequent the Castalon/ Cottonwood area. Javalina are the big concern along the river. Keep food, and things like dish soap and deodorant, safe in a bear box or hard sided vehicle.
With your small trailer, might want to consider a couple nights at one of the vehicle accessible back country sites. Do make a recon with your truck before pulling the trailer down one of those dirt roads.
