Do not go to a laundromat but do go to a service where they clean and iron the shirts and return them to you in the hotel or keep them in the shop for you to collect.
That is what people do who do not stay in hotels with laundry services or who do not want to spend their time cleaning and ironing their shirts.
This service can be found at some laundromats,...
Summit County, CO has a nice, free bus system (https://summitcountyco.gov/586/Transit-Summit-Stage) that runs between the towns and various ski resorts in the area. With just a quick check, I think Arapahoe Basin to Copper Mountain is the longest distance at just under 29km as the crow flies.
In the USA, most hotels above the very lowest-price have an ironing board and iron in each room, or will loan you one on request. I have hand-washed many shirts in the room's sink, left them to drip until mostly dry, then ironed them myself. I did the same in Germany once, now that I think about it.