The number of permits can give an idea of how busy a trail and campsites are. For Yosemite, Look at
https://www.nps.gov/yose/planyourvisit/trailheads.htm . Places like May Lakes (25 per day) and Ten Lakes (40 per day) are obviously going to be busy and have quite a few campers and you'll want to avoid them.
Also check out the Full Trailheads Report at
https://www.nps.gov/yose/planyourvisit/ ... lheads.pdf . Dates and trailheads listed have the full quota reserved.
Besides the southern access points that AT mentioned, trails in/near Yosemite I've been on with few or no others are the Parker Pass/Koip Pass/Gem Pass trail and the high trail from Lewis Creek to Isberg Pass. In addition, you can find more secluded campsites at places like Bernice Lake if you are willing to travel through a popular area and stretch for something more remote. Or better yet, head for something cross-country. For example, routes out of Devils Postpile are popular. But if you go cross-country to Beck Lakes or Holcomb Lake, you'll likely be the only party camped there.