When you move to a new state, the local state laws typically dictate a certain time period in which you have to establish your domicile there, meaning get your driver's license, car registration...basically anything that will collect money for the state.
Simply having the intent to return to a different state is not sufficient except in certain cases, like you're active duty military or you're a student in that state for the explicit purpose of attending a school. To my knowledge, no exception exists for completing GME, but if you really feel strongly about keeping your old license and registration, then you could check on it. I would imagine that it would be a tough sell, however, since residency is a job, so you're not considered a student, and your institution will blink at you if you tell them to withhold state income taxes to another state.
There are two risky pathways here:
A) Keep your old driver's license, car registration, etc., but maintain your home and pay income tax in your new state. This is essentially playing both sides of the fence. As others have pointed out, it will probably only become an issue if you're pulled over on an unrelated offense.
B) Try to keep everything in your old state, to include maintaining an address and paying state income tax there. This is definitely not the path of least resistance, and it may not even be legal. In which case, you run the risk that the state where your training is located will eventually come knocking for its state income tax, wondering why you've been there for so long without paying.
Honestly, we're talking about such a small amount of income that it's unlikely anyone will notice. Of course, we tend to be risk-adverse people, so - as already pointed out - the only safe path is to simply switch everything to the state of your training program. Then be prepared to do it all over again when you're done.
Law #8: They can always hurt you more. -The Fat Man