Leverick used to have laundry before the storm but it was discontinued. It may be back but you'd want to check with them. Guavaberry has laundry but I'm pretty sure it's for guests, only. No help for Old Yard. Again, you may want to contact them Not much help but, that's all I know
If on a boat, Bitter End Yacht will do laundry for you. But based on your questions, I am assuming you are staying on land.
Can confirm Leverick does not currently have machines for self-laundry. They are apparently going to be putting them in when they finish the renovation on the building they have been working on forever though.