My employer asked that I not change my address with them when I moved.

I moved from Tennessee to Ohio in April of 2023 and continued my same job but remotely. We use Gusto and after I had updated my address in it, they changed it back and my employer asked for simplicity sake if I wouldn't mind just leaving it as TN and I can update in January of 2024.

As we get closer and closer to the deadline, I'm beginning to worry that I'm going to get dinged because they didn't update it correctly. They're saying it's fine but Ohio has more taxes than TN so I'm not convinced. I don't have extra money lying around to get dinged for hundreds or thousands of dollars. Any thoughts? Thanks!