The short answer is NO, your mail service is only a mail forwarding address and nothing more. It is not your domicile and as long as you vote, have licenses, pay taxes in your domicile state there will be no issues. But I'm a little confused. If you are full-timers you must have a domicile. Why not just use a mail-forwarding service there and avoid the confusion? It does make everything easier.
That would be my answer too. Your mail forwarding service can be just that...a mail service and doesn't necessarily have to be in the state you're domiciled. For example we used a US-based mail service for many years while living (and domiciled) overseas.
BUT as fulltimers with no other links I think it makes sense to combine the two to avoid any confusion. In that case I would advise getting your mail-forwarding service in the spot you intend to domicile. The popular full-time states (FL, TX, SD) all have plenty of competitive choices so I don't see any reason not to do it.