They have other ports on the Black Sea, the only sort Georgia has (except on the Caspian Sea, but Russia has ports there as well.)
There are several ports or port sites on the Russian coast of the Black Sea. Not more than 150-200 miles up the coast, and only about .70 times that much farther north.
No, it the oil pipeline, and the Old Empire, that they want.
If they wanted a better warm water port, they'd have to invade Turkey, or Iran.
I am beginning to think they just want expansion, after reading some of their newspapers.