Do I have to Pay Georgia State income taxes if I work abroad and have military orders as a civilian (contractor)?

Normally yes. Read the residency requirements of Georgia state. The state may consider you work abroad as temporary and transitory (even if it is for a couple of years).
You foreign income is also subject to SE tax at 15.3% (just as if you are in the U.S.). On federal tax return, you can get credit for taxes paid in the foreign country but this credit is not available on the state tax return.
Read: Your U.S. Tax Return: U.S. Citizen or Resident with Foreign Income

This is especially true if you leave family behind in Georgia.

However, if you have no family, you MAY be able to establish residency in Florida or Tennessee (neither have a state income tax) before departing, thus saving you from paying Georgia income tax.

However, when you return, you had better live in that state for at least a year. If you immediately return to Georgia, and the state figures out what you did, they will DISALLOW the move and charge you state income tax retroactive to the date you left.