Well, if you have corks popping out, you have to do something. But, if that's stopped, then I'd leave it be. Yes, you have some sediment in your bottles, but you can always decant to serve the mead. That will also allow the carbonation to come out of solution.
If you really, really, want to rebottle, you can. I'd open them, and every so gently siphon them to a bottling bucket in which you've dissolved one campden tablet per gallon of mead in a little water. Then rebottle. But if it was up to me, I'd leave it. Just drink it, and realize that experience is the best teacher of all.