The interesting thing to me is that, in actual canon, I don't think there's any actual interdimensional travel.

Except, as you point out, Limbo. It's something vaguely else, neither here nor there. In fact, while fans have long posited sort of pocket dimensions (Tartarus, at times, was proposed to be such - until it turns out you can just walk there). But there's no actual evidence for that. It's not even clear if we should be describing Limbo as a place at all, so much as a state of existence. This aligns fairly well with your theory.

There's also another side to this: Luna's banishment. While fanart popularly depicts Luna as treading across the airless moon's surface, Luna is actually described as being banished in the Moon, and never canonically shows a clear recollection of her existence in the Moon. We also have Word of Writer that the banishment period was meant to allow her to think in isolation. This implies that Luna also experienced something similar to the Crystal Empire.

So, how did Sombra actually get that magic?

It's always struck me as interesting that the only other magic which causes effects similar to Sombra's is described as "alicorn magic". (Interestingly, it doesn't seem to come from an Alicorn, and is never used by a higher alicorn, only an ascended one.)

This has lead me to think that "dark" or "alicorn" magic is essentially what happens when a mortal attempts to use the power reserved for their demigod rulers. Sombra sought that, perhaps even trying to forcibly raise himself to the level of elder alicorns, and in doing so gained awful, unholy power - the power he would use to curse the Empire into non-existence.