I owe more tax to the Federal Government, but will be receiving a Georgia State refund. The way I see I can do that is to file an Extension on the Federal Taxes I owe.

No, filing an extension for your Federal Tax Return does not extend the time you have to pay the tax you owe.

If you are unable to pay your total balance due by April 15, 2019, you should go ahead and file your return on time and pay the balance as soon as possible. 

You also have the option to set up a payment plan with the IRS.  You can do this through TurboTax or directly online with the IRS. 

By filing on time, you will avoid a late filing penalty.  By setting up a payment plan with the IRS, you will avoid accruing additional interest and penalties. 

Here are some links to help you through the process:   

•       How do I set up a Payment Plan with the IRS?

•      https://www.irs.gov/payments/payment-plans-installment-agreements