My state withholdings not reflected in W2

I as remote colleague moved from one state to another (no state income tax) mid year and promptly notified my employer. it took them a month to process it - in the mean time there were two additional paystubs with state taxes paid to former state. However when the W2 showed up it shows a state withholding as of the day I reported (two less payments). Should I contact employer about this or can this be done through tax filing