Welcome Karen, I think a mobile dog grooming gig would work well for a mobile life style, if as you said you don't need a license in the area you want to go to.
Seems you already have your ducks in a row, but how bout checking with your current customers if you can use some of them as a reference before you hit the road.
When your in an area maybe Veterinarians would let you leave your card them them if they don't do grooming, seems you're very well established.
Just had my dog groomed yesterday, and started thinking about mobile groomers for the first time, don't know why it's taken me this long to think of it, guess I'm slow. But worry about how my dog is being treated more then the cut. The cut grows out. I trust our current groomer but we just had a groomer in this area arrested for animal abuse.