The spot I’ve been to in the White Mountain Wilderness on seven occasions is the South Fork Rio Bonito. It’s a little north of Ruidoso, just past Bonito Lake. If you park at South Fork campground there are four different routes you can take to the top and 25 miles of crest trail. So there are lots of loop opportunities.
There are lots of creeks and springs. The driest part of the year is May, when some of the springs go dry and there may be fire restrictions. There is a good spring at 10,000’ near the intersection of the crest trail, South Fork trail and Bluefront trail that allows you to camp on the ridge. There’s a reliable spring in the Ski Apache ski area called Ice Spring on my map.
I’ve never seen a crowd a mile from the road; I’ve always had my choice of campsites. The canyons to the west, Turkey and Argentina Canyons, get quite a bit of horse traffic, but much less on the South Fork.
I really love the thick forest full of elk, the bald ridges with awesome views, and the chance to climb a real mountain, Sierra Blanca. I’m surprised you guys haven’t been there.
The Destinations Southwest forum on Backpacker.com has had a lot about it. There’s also a review here:
http://www.thebackpacker.com/trails/nm/trail_468.php
Call the ranger and get a map!