Company didn't change my address when requested

We spent the entire tax year living in one state, however my company didn't change my address in their system as requested until a couple months later, so for a few months I paid taxes to the previous state and underpaid taxes to the new state. How do I enter everything such that I can recoup taxes from the former state and pay back the underpaid taxes from the first few months of the year? My company has pushed back on issuing a corrected W-2, but do I need to push this with them?