Being June, you'll have lots of hours of daylight so you won't need lights very much. If you get cold easily, bring an extra blanket. You shouldn't need the furnace.
As far as showers, one of the first places I always think of in a town is the community pool. Even though we have a shower, we still go to a pool once in a while for a loooong, warm shower. Usually fairly cheap plus I can get a swim in as well. Unfortunately, the Estes Park Pool is closed all summer for a remodel :(
There used to be a couple of laudromats in EP that also had showers. Can't vouch for them as it's been too many years.