Not sure about Natural Bridges towing a trailer as we have not been there in a long time and I don't remember the roads; should be fine though if that is what you want to do.
Hovenweep is a ways out but we were there again a couple of years ago and they have really made this a great, great place to visit. Good roads if towing.
Moki Dugway is no place for an RV - have driven it a few times. It is a winding back and forth shelf road with some tight turns and serious dropoffs.
Another alternative place to stay is Cadillac Ranch in Bluff. Nothing fancy but a good place. And a great centering location.
You should drive thru Valley of the Gods as well; beautiful drive and a lot like Monument Valley.
Enjoy all these great places.
(Edit for typo correction)