Take highway 163, not 261. The dugway isn't something to mess with towing anything. Get a site at Goosenecks State Park or Natural Bridges and unhitch if you want to drive the dugway. YouTube has lots of videos.
There's a 50s style burger joint in Hanksville, open seasonally.
Crossing the Colorado River is a bit steep and winding, go slow and soak in the awesome views. There's a scenic pull off just west of the river, it's amazing and sad to see how low the Colorado is compared to where it was decades ago.
Before leaving for monument valley, there's Valley of the Gods to check out if you have four wheel drive. It's federal land and boondocking is permitted, if your trailer has enough clearance for the trail.
Canyons of the Ancients isn't far if you have time. I've missed it three times going through there, always seem to be behind schedule.
2013 KZ Sportsmen Classic 200, 20 ft TT
2020 RAM 1500, 5.7 4x4, 8 speed