I can't advise you on the pre-I40 route. But once you get on I40 stay on it thru Dallas-Ft Worth. There is a bypass around it but I don't know if you are up to advanced routing(heh). On the other side of Dallas pick up US 287 and stay on that to Amarillo. Then North on 287 to Dumas TX, not that far, and turn west on US 87 to you hit I-25. Take that north to Walsenburg CO then west on US 160 thru some spectacular mountain country. That will get you to Cortez CO in the west of Colorado and that is the edge of the Four Corner's park loop. And right at Mesa Verde NP if you plan to stop there.
Tho you say you aren't interested in sight seeing before the Utah parks I'd suggest a quick drive up Capulin Volcano park which you go right past and can drive up to the top of the volcano in an an hour or so. That's just north of hwy 87 in New Mexico. And especially if you have kids a diversion to Great Sand Dunes NP which you drive near and is exceptionally popular with the younger set. Maybe a half day there and back to Hwy 160 depending on how much time you spend on the dunes.
