I am being treated for undifferentiated connective tissue disease which can be debilitating during flares. Is it likely I will stay in this limbo diagnosis forever?

Possible. As you probably now by now, this condition waxes and wanes with no rhyme or reason. Sometimes, however, patients enjoy long periods of remission which is quite possible to be the case here although no body can foretell.
No' Mctd should not be this debilitating. Other problems may exist. I would get a second opinion.
It's possible. Some patients with undifferentiated mixed connective tissue disease will progress to a more specific diagnosis. However this may not be a good thing.