At your income level, and with your social security already maxed out, the taxes due on your side income (on top of the regular wages) will likely be 22% or 24% (depending on your tax bracket, filing status, and other income) plus 3% for the medicare portion of social security. So 25-30% (of net income after any expenses) is probably a good estimate. You can make payments electronically at www.irs.gov/payments. If you overpay, that comes back as your refund, of course.
Yes, you should plan to make separate estimated payments to your state, you will need to look at your state tax brackets, it could be anywhere from 3% to 13% depending on the state. There is probably an electronic payment web site you can look up.
At this point you should make your Sept 15 estimated payment now, for income earned from June 1-Aug 31, and plan to make a payment by Jan 15 for income earned from Sept 1-Dec 31. Paying now will reduce any interest that might be due to late payments. There are also various ways to ask for an abatement of the penalty if one is assessed.
