We visited part of that area in September 2015 on a trip to Albuquerque and Phoenix. Went to the Grand Canyon in between and took the scenic route via Four Corners and Monument Valley. I highly recommend swinging north into southeast Utah after you leave Farmington. Our route took us west on US-64 to Teec Nos Pos just over the Arizona line, then a right turn onto US-160 up to Four Corners.
Four Corners itself is a bit of a rip-off in terms of the price, although I'm glad we paid it just because it's something unique to see. If you are by yourself it would be less interesting unless you're comfortable giving your camera to a stranger because the whole point of going there is to get a picture of yourself standing in four states at once. Using a tripod and a remote might be tricky just because there are enough people around that some of them might not wait for you, although that would be a much better option than those stupid cell-phone stick things.
From Four Corners we made a left and crossed into Colorado, then made another left onto CO-41, which then becomes UT-162 after crossing the state line. It ends at Bluff, where we made a left onto US-191 to head southwest to the Valley of the Gods (it becomes US-163 southwest of Bluff). This is a very scenic road, well worth driving. We stopped for the night at the Valley of the Gods Bed and Breakfast, which is located near the bottom of the Moki Dugway. Beautiful place, utterly unexpected find in such a remote area (I contacted them months in advance....if you're going in a few weeks, they may not have anything because it is a small place). We drove the half hour back to Bluff for dinner and then came back to the B&B. The night sky was UTTERLY MAGNIFICENT there, really makes you see what we miss out on when we live in urban or suburban areas with all the light pollution. The following morning after we checked out, we went up the Moki Dugway and drove down the dirt road to Muley Point. Great photo spot. We then drove back down the Dugway and back to US-163 and made a right towards Mexican Hat. The road then takes you through Monument Valley. The one thing we didn't get to do was to drive the 17-mile dirt road through the Valley of the Gods. We just didn't have time: the 17-mile drive takes an hour and on the day we arrived, darkness was coming too soon to allow for it, and on the next day we wanted to haul the four or five hours to the Grand Canyon well prior to sunset (bearing in mind most of Arizona does not observe DST, too).
It is the scenery along this route that makes it worth the longer drive, even if you don't stop for the night anywhere. I assume you've seen photos of Monument Valley or else you've seen it in movies (to name but two, Forrest Gump ended his cross-country run near milepost 13 in Utah on US-163 and there is a sign off to one side of the road; also the cavalry scene in Back to the Future III was filmed there). Seeing it for real is that much more incredible.
After Monument Valley, US-163 runs down to Kayenta and puts you back onto US-160 over to AZ-98 as you've described.
I haven't measured out the comparative distances, but I assume I've given you enough detail to allow you to do so if you want. I can post some pictures from US-163 and maybe UT-162 if you'd like, especially if I wind up having tomorrow off work. Let me know.
