There are two places that I know of that offer showers for a fee. The first, The Pinedale Aquatic Center, I discovered from reading a trip report about one year ago; I have no first hand knowledge. I presume the daily admission fee of $5 covers the shower. See http://www.pinedaleaquatic.com/#!hours-fees/c1kr9. The other facility is an old campground that has seen better days. My last visit was two years ago at which time they charged $5 for a shower. There is almost never anyone on duty there. A self-serve box is near the front door. I am having a little trouble finding the name/address but I believe it is the Pinedale Campground on Jackson Ave. about one block south of the main highway.
I also would like to find an inexpensive place to stay in Pinedale. Last year I paid about $100 for a room at one of the fancier newer lodges at the west end of town. Two years ago I paid about $85 IIRC at the motel in Boulder, about 10 miles south of town.
Looking forward to others' responses.
Edit: minor error