It would be helpful on your posts if you would create a signature line with the size and type of RV you have.
If you have a smaller RV, Mather or Desert View campgrounds in the park are very nice if you can do without hookups.
Since you said Camper Village in Tusayan is only accepting walk-ins, I'd try for that if you can stay in the Flagstaff area the previous night so you could drive to Camper Village in the morning when you could probably get a site when folks are leaving.
In the meantime, keep trying Trailer Village in the park itself as it is definitely the best place to stay. There are always cancellations. The free shuttle stops there and it's even walking distance to the canyon rim.
Otherwise, Williams is the closest you'll be able to find a RV park and that's 50 miles away.
Full-Timed for 16 Years
.... Back in S&B Again
Traveled 8 yr in a 40' 2004 Newmar Dutch Star Motorhome
& 8 yr in a 33' Travel Supreme 5th Wheel