Your Washington license is good in Washington, but likely does not transfer to another state that requires licensing. The certification represented by the license might be a basis for getting licenses in other states, but documentation of your training will likely be more useful. You might also expect background checks.
Most states will want you to have a sales tax license, particularly if services are taxed. Some will want a business license. Many local governments also require a business license for any type of business, may have specific licenses for your business, and might have bonding or liability insurance requirements.
That being said, enforcement of licensing and tax codes is difficult. Tens of thousands of people in this country, both permanently located and itinerant, operate outside the legal structure offering short-term small services like lawn care or gardening, property maintenance, technology help service, and personal services like yours. Other occupations include entertainment and selling handicrafts, although the legal system often has some accommodation for these sorts of business at special events.
I think most people in these gray areas of business understand the risks. If caught, they might pay a fine, do some short jail time, negotiate a solution by agreeing to desist or get appropriate licenses, or just move on. In this community much of this activity is policed by Neighborhood Services, which might issue a citation similar in effect to a traffic violation, or warn you to cease, have you appear before an administrative judge. But I've also seen cases locally go to the municipal court or circuit court as misdemeanors, and your situation as a single parent of small children is precarious: you could temporarily lose custody of your children if arrested on suspicion, however briefly.
Whether a RV park or a public campground would allow you to operate your business might depend first on the legal situation, and after that how your business fits in with their business.
Tom Test
Itasca Spirit 29B