I would plan on going the southern route (I-40). I-70 and north can get quite snowy in late March (some of our best skiing time!)
CloudDriver mentioned Sunset Crater and Wupatki NM's. Also in that area is Walnut Canyon NM. Definitely worth walking the stairs to the bottom of the canyon to see the ruins up close. Look closely and you can still see where someone's fingers applied the mud to the chinking between the sandstone blocks. Of course all three of these NM's are near Flagstaff, which at 7000' can still be a little wintery in late March. (I have to assume late March is spring break 'cause otherwise a little later is better for this relatively high area -- the south rim is also nearly 7000' in elevation and snow is not uncommon in late March). While not as big as Mesa Verde and some of the other ruins of the SW, both Wupatki and Walnut Canyon will give your kids a good exposure to the Native American culture and structures of the southwest.
I'd suggest Trailer Village at the south rim so you have quick access to the rim. (The shuttle typically isn't running at that time of year.) Head out there to watch the sunset at least one night. You'll know you're not in Indiana anymore!
When we travel in late fall and early spring and will be camped with electricity, we take an electric heater with us (the kind that look like an old radiator). Quieter at night and besides, why burn the propane if you have electricity? Prepare for cool temps. The average high for late March is mid 50's and the average low is in the mid to upper 20's.