I think the solution is related to modal logic.

Interesting idea. But what is it that shifts us into a new universe? A clause of the form "___ is true"? The use of an indicative "this"? I like the idea of a universe disconnected from the rest of reality. But what puts us there, and what can we talk about while in residence?

You might enjoy Vicious Circles which sketches a resolution of the Liar which seems similar to what you are suggesting. Your idea may also be very similar to the "relevance logic" and "paraconsistency" approach sketched in the article linked by the OP.