I doubt you'll find an apartment with a washer or dryer in either location but you can get it done for you very cheaply on Utila and probably Roatan. I rented a two bedroom apartment without air conditioning on Utila for $400 a month that I shared with another person making it pretty cheap.
I have a number of apartments i rent to Pro candidates while they train with me here in Roatan. Two bedroom shared apartments range from $250/month with internet, washing machine and in some cases AC. For a 1 bedroom (not studio) with AC, Internet and washing machine you will pay $400.
HTH
Will