Will I be able to enter Turkey as planned
No, definitely not as planned. You might be allowed to enter (I doubt it) but can expect at the very least to have to provide quite a lot of clarification. To be directed to the wrong gate is already very unusual, for you then to accept that misdirection makes the chance of being in a situation that happens quite often (for which there may be special procedures in place) extremely slight. Signs in IST are visible, of a high standard, and bilingual:
Image courtesy Mike at upgrd
or am I unable since I have both entrance and exit stamps on my visa?
Technically, you entered Turkey once the ink from the entry stamp hit your passport. That is your single entry used up, with or without an exit stamp. With an exit stamp on the same day showing the same location and stamp number may however help with any 'negotiation' required with a consulate.
Given the significance for you, I suggest you contact the Turkish Embassy in Tbilisi as soon as possible:
Telephone: 00 995 32 225 20 72 -73-74-75-76
I would appreciate it you would revert here in due course with the (hopefully happy!) outcome.