I don't know how you "apply" to become a resident of a state. Whether you are a resident is determined by the facts of your situation. It's not something you can apply for. Generally you become a resident of a state when you move there, but the rules can get complicated when the situation is not clear cut.
Different states have different definitions of a resident for income tax purposes, which may be different from the definitions for other purposes. Because the states have different definitions, it is possible to be considered a resident of more than one state at the same time.
How you handle your taxes will depend on exactly when you become a North Carolina resident for income tax purposes, when you stop being a Pennsylvania resident for income tax purposes, and when you stop working in Pennsylvania. Those could be two or three different dates.
For a summary of Pennsylvania's definition of a resident for income tax purposes, see "Residency Status" on pages 2 - 6 of the Brief Overview and Filing Requirements in the Pennsylvania Personal Income Tax Guide. You will see the terms "domicile" and "residence." You can have only one domicile at a time, but you can be a resident of more than one state. Note that Pennsylvania considers you a resident if your domicile is in Pennsylvania OR if you are a "statutory resident." You are a statutory resident if you have a home in Pennsylvania and spend more than 183 days of the year in the state. So in the year that you move to South Carolina, you could still be a Pennsylvania resident for income tax purposes.
Unlike most states, South Carolina does not make a clear distinction between domicile and residence. Here is a link to a guide from the South Carolina Department of Revenue that discusses how to determine whether South Carolina is your domicile or residence for income tax purposes.
