I always do the laundry at WDW, it is just something that is necessary since we usually take long trips and can't pack enough clothes for 10-12 days.
One way I do this is by taking along some clean white garbage bags. I make sure everyone puts dirty whites in one bag and dirty colors in another, no sorting in the Laundry Room. I also get a box of those laundry tablets at the grocery store with indivually wrapped detergent and a small box of softener sheets so that I don't have to purchase these at the resort. Also on my packing list is a small roll of quarters for the machines. When we are getting "low" on clothes I just put everything in a rolling suitcase and head off the the LR.
I have found that early morning (really early at the value resorts) is the quietest time, but it all depends on who needs laundry done. One time I work up at 4am & couldn't fall back asleep and met 2 other moms in the LR who said the same thing! If you are at the pool or back at the resort and know that you'll have a couple of hours downtime, head out and stick some laundry in, you'd be surprised at how little time it takes. I have also done it in the early and late evening, and sometimes the room is packed, sometimes it is empty.
I realize that you are not staying at a DVC resort, but if you do then the laundry facilites are free, otherwise it will cost $2 for a wash and $2/30 minutes of drying time. Obviously these are commerical type dryers you'll see at a laundromat and tend to get pretty hot so watch out for those clothes that might shrink!
If you send your things out be prepared to pay about:
$2 per pair of socks or underwear
$3 child's top or bottom
$4 adult t-shirt or lightweight shirt
$5 adult pants
$5+ for jeans, sweater/sweatshirts, dresses, slacks, etc...
I remember seeing the list and thinking it was pretty outrageous!
Anyway, if you are willing to invest a bit of time and maybe roping DH into helping you can save a lot of $$$!