I agree with others, if you can get the diapers down there, try to do that. That's what we did when we traveled and DD was still in diapers. We always rented a car so it was easy.
I've used the laundry facilites at the All Star Movie resort. It was right near the pool (if I remember correctly - it was a while ago). We were there for 2 weeks and there was no way I was packing 2 weeks worth of clothes so I did laundry mid trip. I think it was $2 per load to wash and close to that, if not the same to dry. I think I consolidated and did just 2 loads, darks/denim and lights/whites. Did everything at the same time because there were lots of machines. It wasn't a big deal.
Gail
80 & 91 WDW OffSite; 10/95 DL Honeymoon; 4/00 & 5/01 DL Off Site; 12/02 WDW - ASMo;
10/03 WDW - CSR; 5/05 WDW - ASMo; 2/07 WDW - SSR; 8/07 WDW - Pop - Me & DD;
1/08 WDW - CBR - DH & Me; 8/09 WDW - THV & SSR; 08/11 WDW - Pop w/Free Dining