I would agree with dry cleaning.
There is a laundromat in Puerto Plata as well ( maybe more that I dont know about ), close to Jose Luis Supermercado. I dont know how busy it is.
And as you say, if you have money you have your own washingmashine, if not you borrow, hire or wash by hand.
If you live a normal life here, have employment etc, it is also common you just have someone who does the cleaning and laundry for you, a maid coming every week, or the neighbour etc.
Smaller hotels usually have a washingmashine or 2 on the property for guests to use. A washingmashine for 5-10 000 is not a big investments for most hotels, as they still need it for their own laundry..
I would look into and learn more about the drycleaning possibilites if I were you.
Good luck, and let us know about your plans!