Who knows, but the official wording was:
There is no mention of not progressing, so the obvious answer is, there will be no option to stay.
Who knows though? Maybe they will allow some sort of anniversary era realms (with dual spec). Or have a way to go from Vanilla Azeroth to TBC through Anniversary HC realms going forward, since there’s also no mention of what will happen there, but they leave out the Anniversary HC realm from progressing to TBC. So, what happens if you die after the fact? That’s an interesting situation. Vanilla HC to TBC Normal server?
There are some differences in how Anniversary and Era servers work, and of course dual spec doesn’t exist in Era.
We also don’t know what’s going to happen to SoD characters or what happens when TBC content dries up.