I tried adding a cap to ground off the wiper. That seemed to help a tiny bit, but not much.

I had the same idea about sampling and averaging in the shower this morning. Unfortunately, I haven't had a chance to try it yet. I'm using a 16F676 which only has 1023 words of program memory, and I've already been "trimming the fat" to get what I've already got working on there. I need to use a PDIP14, so my options are limited. I just ordered some 16F616, so that should give me twice the memory.