How do I stay on top of payroll taxes and filings?

You can run payroll, which is no small thing. What stresses you now are the parts with real consequences: the withholding, the filings, the rules that change from state to state. That stress is reasonable, because these are the pieces where a mistake actually costs something. The good news is that this is a solved problem. You do not have to hold it all in your head or get it right from memory. This is exactly the kind of work that systems and specialists exist to carry, and leaning on them is not a weakness. It is how careful people handle high-stakes routine. So take the piece that scares you most, the multi-state compliance, and either lean harder on software built for it or bring in a professional for that part. Free your attention for the judgment calls only you can make. You do not earn points for carrying alone what a good tool can carry for you.