Hi there, looks like we'll be sailing together. The Spirit does not have a self service laundry. So you have three choices, either pack enough clothes for the whole trip, pay the big bucks to have them washed on board by your room steward or find a place in one of the ports that will wash them.
On another board it listed a laundry in Aruba called Madamme Janettes that will do your laundry for the cost of the machines and a tip. The directions I got were: it is on the right hand side of the road from Adventure Golf to the restaurant. It is too far to walk and you'll need to take a taxi to get there. so we thought if we feel like we may run short of clothes before we get to NYC, we may just get a cab and drop off our laundry on our way to the beach, then pick it up on our way back to the ship.
I will message you the details with the info on a meet and greet on board and hope you will join us!