There are two places in Truckee, California where travelers can shower without getting a room. There might be others.
The first is the Community Recreation Center. Open Mondays through Fridays, 6AM to 9PM, Saturdays 8AM to 1PM. It's a very nice place. For $5 you can use the gym and the shower. There's no shower-only option. The drawback is that there's just one shower. You might need to wait your turn. Bring your own soap and towel.
The other is the Community Swimming Pool. It's behind the high school. Go around the left side of the building and follow the fire lane to the back. For $5 you can swim laps and shower. A shower by itself is only $2. The problem is, pool hours are all over the map. So, while it's cheaper, it's harder to fit into one's schedule. That's why I ended up at the rec center. And it's why I don't know about soap or towels at the pool. But I'm squeaky clean.