I efiled my federal and state returns yesterday, paying the tax for each. Do I have to follow up with mailing paper copies, with schedules and forms, like 1099-INT?

If you e-filed your return, you do not have to mail any paperwork, unless the program instructs you to do so. It has all been done electronically. Your filing instructions will tell you if you need to mail anything to the federal or the state. Just save your W-2’s and other paperwork for your own records.  You should also print a copy of your return for your own records.