I'll give some general ideas to help you get started on solutions that would work for you. We never actually full-timed - we are "long timers", several months a year, but the situation is much the same.
MAIL: These days you don't need to get much US mail - nearly everything can be handled electronically, email, text messages, fax via PC, and direct file uploads. That includes online banking and bill pay. For the rest, establish an account with a mail-forwarding service and use their address as your physical mail address. These services can either forward mail to you (in bulk) at whatever location you specify, e.g. a USPS General Delivery PO, or open and scan mail to send electronically. There are a variety of services and way to handle the mail.
INSURANCE (Etc): You still have a legal "home" somewhere, called a Domicile. As a fulltimer, you have some latitude as to where that is, but it is somewhat determined by where you choose to make your legal relationships, e.g. driver license, vehicle tags, voter registration, etc. Health insurers generally accept the domicile location as your "home" for insurance purposes. However, since you are often "away from home", you need a policy that has a broad network of providers and does not penalize for out of area services. Therefore most HMOs are to be avoided, as are state or regional providers. United Healthcare, Aetna, Anthem and other major insurers offer nationwide coverage in most areas, but the availability of those policies is one of the factors in choosing a domicile states.
Much has been written here about the concept of domicile, so I suggest using SEARCH to find previous conversations about it. Unfortunately, it's a somewhat hazy concept when it comes to how you establish domicile.
HOUSE: As long-timers, we of course kept our house, essentially our winter home (in Florida, in our case). We "closed up the house" just like many others who have two homes do, shutting down some services (e.g. cable tv), forwarding mail and landline phone, shutting off most of the water lines, heat or a/c set to low or high temps, arrange for lawn care, etc. If you expect to be gone long term, renting it out is a possibility as well, but that has both risk and reward.
One advantage of keeping a house is that you have a ready-made permanent physical address to use for the purpose of insurance, icenses, etc. That makes those things somewhat easier to manage.
This probably raises as many questions as it answers, and there are many variations of how to handle each item I've mentioned. Surely others will voice their solutions as well.
