There are multiple spots at both Cottonwood Lakes and Cottonwood Pass trailheads. According to:
https://www.hikespeak.com/campgrounds/horseshoe-meadow-campgrounds-golden-trout-wilderness/Eighteen sites are set just off the road near the Cottonwood Pass trailhead and twelve at the adjacent Cottonwood Lakes trailhead. Both are $6 per night with fire rings, picnic tables, potable water, and vault toilets. These sites are classified as “walk-ins” because campers do not actually set up their tents right by their vehicles. Instead, campers must walk in a couple hundred feet from the road. Additionally, ten equestrian sites designed for human and horse go for $12 a night (prices as of 2018).
I would have guessed a lot more spots than that. I've only stayed in the Cottonwood Pass sites. It's a large area. I've been there twice in the past couple years. Didn't have a problem finding a spot.
