My buddy and I have to make a run down to the Navajo reservation next week to drop some things off at a relatives house,but wanted to maybe make a detour around the four corners area and hit any 5.9-5.11 sport multi-pitch.
We need to get back to Denver eventually, so we wanted to maybe stay on the eastern portion of the four corners (aka the two corners- NM and CO).
You'll find relatively little multi-pitch sport climbing in NM. The area most known for this style in NM is Diablo Canyon, near Santa Fe. A few other multi-pitch sport climbs can be found in the state, in the Sandias, Gilman Tunnels, and Bat Cave (T or C).
Though perhaps prohibitively far from where you're going, the place that immediately comes to mind as being known for multi-pitch sport is Cochise Stronghold (southern Arizona).
Other options (not counting those near Denver) worth looking at on the internet include Snow Canyon (Ut), a few routes in Zion, Maple, some stuff near Ophir & Telluride, and there are probably a few multipitch sport climbs in central Arizona (Sedona, Flag, Prescott, Isolation Canyon).
Sometimes a small amount of trad gear is recommended- carefully check route descriptions.
Western Utah is out of your range? Prophesy wall close to St George has some fun sport multi-pitch. Outside of New Castle CO there is some sport multi up the West Fortress. Buy the Western Sloper Guidebook from Wolverine for beta on that crag.