I recently moved states and continued to work in the old state. What you say lines up pretty well with what I had to do (although since I continued working in the old state, rather than start work in the new state, it was the old state that I filed both Non-Resident and Part Year Resident for). My state in particular (Massachusetts) has a single form for filing as both a Non-Resident and Part Year Resident, although there's an extra worksheet you have to fill out if you're using it for both.
I ended up filing my taxes on my own, but would recommend just getting someone to do it. I normally file my own taxes, but without using any software -- just directly filling out the official forms. The instructions for Non-Resident/Part Year Resident (once again, for Massachusetts at least) were rather vague and hard to follow, even as someone who is fairly used to reading tax form instructions.