The pay shower facility at the RVG store is run by a concessionaire. It and the laundromat should still be available.
I just mentioned the lack of showers in the NPS campgrounds restrooms because that seems to be the main complaint people have.
The water shortage alert is still in force, with visitors 'limited' to 5 gallons per day. The Chisos Basin is so short of water and dry that smoking is prohibited on trails and discouraged elsewhere as a fire danger.
EDIT - I was mistaken. The OP eggshell trailer has a 30 gallon black tank and a 28 gallon grey tank. A two week stay with careful management of water would be possible without dumping even though there is no dump station at/ near Cottonwood. The closest is the one at Chisos Basin.
The NPS restrooms do have a dish washing sink. One can use those or a tub in the trailer for dish washing and dump the water in the restroom sink to save tank capacity.
It will be a 50 mile drive / over one hour to the RVG pay showers, and likely a wait for one of the three showers to be available.