Help finding nightly parking for our caravan

Most campgrounds in the mountain west will be closed at that time of the year. You may be able to smile and dial and find some parks in the southwest that will have space since the snowbirds will have not yet flown in large flocks that direction. Appeal to greed by offering to rent multiple sites and offer to not leave until those sites have been inspected for cleanliness. Truck stps would be a horrible idea. They are almost always packed to the gills and a trucker finding multiple parking lanes taken with dog runs might just flip out. Best option might be to call towns along your route and inquire about possible parking at fairgrounds, rodeo grounds or other public places. Once again, the profferring of monetary compensation could grease the wheels.
I agree with another poster in that you should have your accommodations 100% secured before you depart. Much better to spend a day on the phone than it is to spend a night driving around trying to find a place that won't just laugh wgile showing you the exit.