If I had continued as self-employed, I would definitely pay estimated taxes in June, September, and January. But since I switched in May, I'm not sure if I need to do that (i.e. if I'd be overpaying a lot). Do I only need to pay estimated taxes based on the self-employment income (Jan-May)? In that case I'll probably pay the June quarter but not the September and January ones.
You don't have to pay estimated taxes, but you might be penalized for under-withholding if you don't. However, as long as you meet the safe harbor withholding requirements, you're in the clear. The safe harbor is the lesser of: 100% (110% if AGI >$150k) of your previous year's actual tax liability or 90% of the current year's tax liability. So as long as you're meeting that, you can do it all through payroll withholding if you want.
If you selected the default withholding election on your W-4 (withholding certificate), you might be withholding too much or too little. You should be able to adjust your W-4 to get to a reasonable number by the end of the year. The IRS' withholding estimator can help you figure out if you need to change your withholding.
The estimated tax payment option is mainly a way for people who are not employees - and don't have taxes withheld by their employer - to pay taxes in a timely manner and avoid penalty.
Now that you are having taxes withheld, you can completely pay in your obligations that way. In fact, if you were employed and also doing a side gig as a contractor, you could adjust your withholdings to cover both situations.
The W-4 form is the way to make that adjustment, it now includes questions about any side business. Also you can adjust it periodically if it becomes clear that you're over your target amount or under it.
The safe harbor calculation is a guaranteed method of avoiding penalty.
The only way to guarantee that you won't owe or have a refund is to know exactly what your tax bill will be for the year, and pay that amount in (between witholdings and estimated taxes) - hard to do. But the W-4 estimator along with comparisons to the prior year should get you close.