Yes, Spring and Fall are very busy times. Never visit during the week before Easter when the Jeep Safari is in full swing. Lines at the gas pumps, lines at the restaurants, lines to check out at the grocery store that extent to the back of the store.
Talked to one BLM ranger there last fall who said something like 2.4 million people visited the Moab area in a year. Consider that the entire population of Grand County is under 10,000 and Moab is just over 5000 and it's easy to see why the facilities are stretched to their maximum.
Of course I'm part of the problem. I camped there for a total of five weeks last year.