If you are in decent shape then consider hiking down and staying at least a couple nights at the Phantom Ranch. They have cabins and dorm style accommodations.
Folks with backpacking gear can stay at the campground and arrange to have just your meals at the ranch. The meals are served family style in a large dining room.
If you are into fishing, it used to be that National Parks don't require a state fishing license. At certain times of the year the trout fishing is pretty good. The rules about tackle may have changed but you used to be able to use anything except live bait. Now you might have to use barbless hooks, that seems to be the trend.
There are also several nice hikes at the bottom and you'll be away from the crowds at the top. Here's a link to get you started.
Phantom Ranch / Bright Angel Day Hikes
The hike down is pretty easy but go slow if your knees are getting old. The hike out isn't too difficult as long as you start early and don't carry too much stuff. I always stash a few water bottles on the way down.
Another spot with good views of a spectacular canyon is Goosenecks State Park which is just a little West of Mexican Hat UT near the AZ border. You can drive right up the the edge of the canyon and camp if you want to. Use Google satellite view to get an idea of what the area looks like.
There is a good spot to eat in Monticello UT called MD Ranch Cookhouse. It's right on the main road. The folks in that town were extra friendly even though we were a bunch of smelly backpackers. This part of Utah is over 6000 feet elevation so you'll have warm days but chilly nights. It can get down below 40 even in June.
If you want to see some of Utah's famous arches but want to avoid the crowds there is a nice spot called Natural Bridges National Monument a few miles West of Blanding UT. There is a nice loop drive with lots of easy parking if you are inclined to stretch out your legs or take some pictures. If you take that side trip then you don't have to drive all the way back to Blanding, head south to Mexican Hat on 261.
A good spot for stretching out and a nice steak is the Cameron Trading Post. It's in Cameron AZ, about an hour North of Flagstaff at the intersection of 89 and 64. 64 is the back way into the Grand Canyon but the rules about driving at the canyon would be something you would need to check out. I know you aren't interested in tourist traps but there is also a large gift shop where you can see lots of good examples of Navaho and Hopi art and jewelry. My wife is pretty picky about jewelry but I was able to find a nice pair of earings with a SW flavor that she likes.
For a good view of the AZ Painted Desert, especially around sunset, about 20 miles South of Cameron turn West onto forest road 418. You'll know when you get close to the turnoff when there are old volcanic cinder cones on both sides of the road. Just drive a mile or two and stop for a view back to the East. This spot is one of the best to see exactly what the heck a painted desert is.
If you continue on the forest road another few miles it takes you to a spot called Lockett Meadow or The Inner Basin. In the winter I have walked this a few times so it probably isn't too far. Here you'll be in a flat area where you can see into the center of the San Francisco Mountains, at about 8500 feet with groves of aspen and ponderosa pine. Use google maps and click the "Terrain" tab to see the layout.
The Inner Basin is often visited by Native Americans. They come to the mountains to place prayer sticks, I think they are in honor of their ancestors. The sticks are 2 to 3 feet long and are decorated so you won't mistake them for an ordinary stick. I have seen them a few times and you're aren't supposed to mess with them, that would be like messing with a gravestone in a cemetary. Even if you aren't a religious person seeing some of these prayer sticks will be memorable.
