I second MrNoodly's suggestion of the Pilot at exit 185, Bellemont. Friendly people, nice space. They have a good water pump in the parking lot for fresh water, across from the McDonald's drive thru. Some people stay there for extended periods.
Then 15 miles west of there on I-40 is the tourist town of Williams, with a big Love's truck stop where you could check out that great little town and stay at the truck stop. This is another easy-going atmosphere and is pleasant to stay there.
Someone told me there is a truck stop right in Flagstaff that is accommodating. Little America Travel Center. I have not gone there so maybe go see.
Do you know about A-1 Mountain Road on I-40 about 5 or 6 miles west of Flagstaff? It is free camping and very nice. I don't know that you would have a problem with flooding there, and it is close to Flagstaff.