Wow, what an adventure, you are very fortunate that BOTH you and your wife can physically do such a trip, my wife cannot.
My question is, do you plan on returning to your hometown in Idaho when the trip ends? You can't stay out there the rest of your lives, so at some point you have to go and live someplace.
As far as the state in which you came from and have your driver's license in, they don't know you're gone, and there is no reason to tell them. If by chance your license comes up for renewal while you're gone, then renew it online.
If you have family or a very close friend in Idaho, simply use their address as your home address. Of course, you need to ask permission from them, but typically that shouldn't be a problem, you're not doing anything illegal, just using their address as your home address. Then when you go back to Idaho establish a new address once you've found a place to live and get everything transferred.
Mail wise you can get a mail scanning service, RV'rs do this, the postal service scans your mail and you can view the mail online. If you need a particular piece of mail you can get professional RV Mail Forwarding Services, Organizations like Escapees RV Club offer mail forwarding services specifically designed for full-time and part-time RVers. They handle your mail and forward it to your current location. Even though you're on a bike just treat it as if you were in an RV.
Your biggest issue could be filing taxes, but if you do your taxes online, and then use a family members address you'll be ok.
I assume you set up all your bills, if you have any, for paperless filing.