We got our first box style crab trap about 5 years ago, and it had hinging doors. I ran into a similar problem like you have described, we tried leaving it in longer and it helped a little bit. I am rather embarrassed to admit this but we realized we had tied the crab trap upside down. Because the doors were hinged when we would pull the trap up the doors would open, and a lot of the larger crab would scurry out as we pulled the trap up. We probably never would have noticed except we were out one day when the water was uncharacteristically clear and watched it happen from about 15 feet down in the water.
Once we flipped the trap around it started producing as good, if not better than our rings. We typically shoot for the same 15-20 minute soak you had mentioned.