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).
