Try using a diaper rash cream that includes zinc oxide as an ingredient on the rash when you are not running. Especially use it at night. When running use Body Glide or wear half tights. Just wear a top untucked and the half tights won't look much different from running shorts. If you're short like me, cut off the bottom by 2-3 inches. They will be cooler. You can also wet them before or during the run for a cooling effect.
Once the rash heals, you can just use Body Glide... maybe.
If you think it's the seam in the liner, cut out the liner and wear performance boxer briefs instead. I've been doing that for 15 years. You can cut the legs off those as well, but I wouldn't recommend it because it could be your thighs rubbing together that's part of the problem.