We pack with doing in-room laundry in mind-- mostly lightweight, quick-dry clothes, or clothes that will hold up for multiple wears between washes. We bring a spare bar of white soap and wash the delicates and lightweight stuff in the sink for a 7-day, and hang up any "outside" clothes to air out for a day or two between wears. Make sure to wring your clothes out well and let them drip in the shower stall, and once they've stopped dripping, you can move them out to the main room.
If you have a balcony and it's a nice day in port, hang your wet clothes near the balcony door so they'll get some sunlight, or even out on the balcony if you want to risk it (Princess discourages this); otherwise, turn the AC on to a reasonably low temp and your clothes will dry by the time you come back to your room after dinner. Make sure you bring hangers and clothespins for this-- we save the cheap hangers from dry-cleaning and get clothespins from the dollar store, and often leave them behind after a cruise.
For longer cruises, there's no avoiding the laundromat, no matter how much sink laundry you do... we do all the same stuff, except we plan one laundromat trip, knowing that we can't wash jeans/sweatshirts/outerwear in the room. Try going during an early morning or a late night, or maybe the late night after a port day when everyone's tired from their excursions. Every 10-14 day cruise we've ever done has been near or full-capacity pre-Covid, so trying to get a machine took me back to my college days.