Most carriers want their drivers to live within a certain number of miles from a terminal in the state they live in. All large carriers have terminals or drop yards in most states. You don't pay income tax based on the home state of your employer, you pay income tax based on the state you live in.
That is not always accurate. If the state your employer is out of has income tax, or if the majority of your income is earned in a state with income tax, you may be liable for taxes to that state.
So by your reasoning, truck drivers would owe income taxes in every state they drive through if that state has income taxes? Using that same argument, if the state you live in does have income tax, they wouldn't tax you for wages earned in other states? That's not a system I've ever heard of anybody using.
I didn't specify truck drivers, but it is something that oilfield drivers and other jobs deal with every year. In the last decade I've paid taxes to Colorado, ND, and NM without residence in those states due to the amount of earnings generated there. While it doesn't apply to an OTR driver, not all CDL jobs are OTR.
But if I'm not mistaken, don't oilfield workers usually live near the job site for several months a year? My son in law lived in ND for a year working the oil fields and yes he had to file state income taxes there, but that's not how it usually works with OTR and regional trucking companies, which is what I believe the OP was asking about.
It varies, I worked 3 on 1 off most of the time. My first year I lived in Texas, still had to pay CO taxes. Next year I moved to CO due to it being my primary work location, ended up working a large portion of the years in NM and ND (that's a whole other discussion, but both states at the worst time of year to be in them) so had to pay their taxes. My entry into the field was by being a CDL driver though. I'm not saying this is a normal occurrence, but those jobs do exist, and will hire fresh CDL drivers, so it's worth mentioning.
One other that occasionally comes up is living close to a state border and working LTL in an adjacent state. Example is seeing guys from Cheyenne, WY working for some food service companies out of CO.
For the record most of my driving career, I worked for carriers out of my home state. I NEVER had any issues with my taxes NOT being paid to Georgia. Next point. Beware of a recent US Supreme Court decision in State Comptroller of the Treasury v. Wynne, 64 A.3d 453, 457–58 (Md. 2013)! The court held that being forced to pay taxes to two or more states was UNCONSTITUTIONAL! The tax laws can be rather complex. Again I have NEVER had a problem with an out-of-state employer sending in my state taxes to Georgia.
Licenses must only be moved IF you are residing in another state for more than 6 months of the year. IE you have taken a new address in another state and your bills are sent to the new address.