I moved to NY from NC recently and employer is still withholding only for NC, trying to determine what changes need to be made. Thank you.
You need to have your employer update your place of residence for all tax reporting. If you work in a state that you do not reside in you need to make sure that is clear.
Generally you can expect to pay taxes to the state where you reside. You will also be required to pay taxes to the state where you work --none of this is affected by the state your where your employer is located. As a remote employee that spends most of your time out-of-state you would not be expected to pay taxes there.
Under certain conditions you may be required to pay taxes to the state your employer is located in, if you spend a significant amount of time there. (for example some cities levy taxes on visiting sports team players when the team 'visits' and plays a specified number of games/hours there).
If you moved this year, for 2019 you will file as a part-year resident in both NY and NC. NC can tax all your income until the date you become a resident of NY. NY can tax all your income after that point. You became an NY resident for tax purposes on the day you began living in your new home in NY.
Once you become an NY resident, from that point on NC can only tax income you earn from actually (physically) working in NC.
So if you are now an NY resident and you no longer actually work within NC, your earnings (from the date of NY residency) are taxable only by your new home state of NY. So, if that's your situation now, your employer should withhold only NY taxes.
If your employer is unable or unwilling to withhold NY taxes, you will have to begin making quarterly estimated tax payments to NY.