Whilst on the "Dawn" I used on our deck 9 ( I got the impression they were on most decks). There were four washers and four dryers and two ironing boards, these were all free. They had a pay machine for you to get a little box of "Tide" or bleach. I would assume it's the same on the Dream. For swimwear etc, we hung on the line in the shower and they were bone dry in the morning. Kids had gone in the pool one day with their t-shirts on and they dried perfectly too in the shower.
We just returned from a 7 day cruise with the Dream to Cosemel, Cancun, Belize and Roaton, Hondarous. There are no self-serve washers and dryers on the Dream. They do offer laundry service, which is pretty expensive. But, they did offer a laundry special for 2-3 days towards the end of the cruise. Fill one large paper bag with as much laundry as you can for $14.95 or two for $19.95. We took advantage of that. We stuffed tons of clothes in there. It was quite a bargain and we took home some clean clothes.
The laundry promos of $20 for a bag of laundry on some ships is at the ship's discretion! Some times they do it and some times they don't. I wouldn't count on it!
I did count on it and they didn't do it our cruise week so we were somewhat short on clothes!!!