March in Moab isn't going to be empty. The jeep safari is 4/4-4/12 this year just be aware. As some mentioned along hwy 128 and the river, north of town on the west side of the hwy are some areas, not much south of town until you get a ways. One campground while I haven't stayed there recently I used to love it as it was quiet and off the beaten path but easy to get to. South of Moab about 30ish miles is the canyon rims recreation area. Windwhistle campground is a campground and not true boondocking but no hookups. Water and toilets may be available depending on when they open them up. If you continue on that road there are several turnoffs to the south for real boon docking and there are some excellent sites back up in there. Just hit dinner before you head out of Moab or head down to Monticello as another option.
Also Someone mentioned Hamburger rocks down by the Needles district just outside the park. Nothing wrong with the area at all but that is about 70 miles from Moab though you are only a couple miles at most from the needles district of canyonlands np. Also I think not sure if I would call it quiet but kane creek road (at the macdonalds) leads to hurrah pass and on the way I believe there is some boon docking and overflow camping as well but the boondocking is literally right off the side of the road. While no real issue at night during the day the road can have some traffic on it and after a while it turns dirt so can get dusty with the traffic.
