You're allowing $1200/m for camp fees. That's way too much, even for California if you don't stay near the prime areas. Folks that are stating $50/night are in 'resort' parks or near main attractions.
You can easily find RV parks with a monthly rate of $350-$400+ and seasonal rates would be much cheaper.
You mentioned New Mexico...the state parks offer an annual permit for $225/year for dry camping and an addt'l $4/night if you want electric. They will have a dump station available and there are plenty of parks in cooler temps for spring-fall. Winter is too cold for NM.
You asked how long could you go without dumping. That depends on your RV and the holding tank capacities. Smaller RVs will have smaller capacity. We can go 11-14 days w/o dumping but we have large tanks.
I don't think you'd be any more clautophobic in a Class C than you would in a trailer or 5th wheel, especially if you have a slideout. You'd adjust!