I don't know specifically what is wrong, but, yes, your turn signal indicators should show up on your dash as a blinking green light. Since the problem occurred after you washed your machine, it's natural to think water got into something. It is also possible that you broke a connection somewhere. Start by letting a fan blow gently into the dash area over night; that may dry out any water. Unfortunately, you may have permanently damaged something too, and that won't be cured by drying. A trip to the dealer would be necessary. Not much help, I'm afraid, but it is a place to start.
BS