Okay, so staying on-site isn't an option due to the budget-unfriendliness and the unavailability.
It looks like our best bet is it to stay somewhere local but off-site like Williams (I'm just repeating what someone wrote) that first night, driving to the West Rim where it's less crowded (is this good advice?) and spending a chunk of time/half day there. Then late afternoon, heading out to drive a bit more (5-6 hours) to our next destination (Albuquerque?).

What do you all think of this plan? If we wake up very early and get to the western rim at an early hour, will we have a better chance at parking and not be totally crowded?