Company changed my state location from PA to NC too early

I took a new role within my current company at the start of 2021. I was working/living in PA up until that time. Once I accepted the new role, I was still living/working in PA until mid March 2021 since I was still doing my old job too since it was a soft transition to the new role that was to be located in NC. However my company changed my state to NC for my 1/22/2021 paystub and beyond. 

 

I can back calculate using paystubs to accurately determine how much tax should have been paid to PA state and local from 1/1/2021-3/13/2021 (date I moved to NC and worked in that role full time). Is that how I am suppose to file, calculate/pay PA state and local for that time frame and get a refund from NC for that range? Thanks for the help!