I think it is reasonable to assume that Sauron would have become her servant, because she would be so powerful. But he would not do so willingly, and would, without doubt, know enough about the ring to manipulate Galadriel, slowly but definitively. Sauron's existence would have weakened Galadriel, so she might have decided to destroy him.
I think the point is that she was already very powerful. The ring would just have focused that power, strengthened it and corrupted her. And, in the end, she would be the servant of the ring, which, I surmise, would draw power from her too.
The result, whatever, would be destruction on a scale not envisaged in the books. I suspect that, in the longer term, it would not really matter who controlled the ring, the end result would be the same. The only difference is the timescale.
 Following @Richards comment, I think Galadriel's rule would be very different from Sauron's. She would rule many others, all those who had rings. She would control the other ring holders and reign over middle earth through them. It would probably seem like a benign dictatorship for a while, until people crossed her.