Isn't there something in the lore that calls the post-ToR galaxy 'The Dark Age of the Republic' or some such sunshine and roses title?
Yes, but that's at the end of the New Sith Wars, which will begin in about 1500 years, and end in about 2500 years.
Very little of the history between now and then is recorded. This topic comes up again fairly regularly in these forums, and the take-away is simple: we do not know. We do not know what happens to this Sith Empire, but they'll be gone in 1500 years. We do not know what happens to this version of the Republic, but something calling itself the Galactic Republic has existed for 22,000 years and continues to exist for another 3500 years, until the Galactic Empire does away with a 25,000 year old government. Ouchies.
Anyhow, we don't know who wins this war. People make all sorts of speculations and often present them as truths, but they're simply opinions. We do not know how the Emperor dies. We do not in fact know what happens to any of the major characters alive now. We do not know what becomes of the war that restarted at the end of Ch 2. We don't know how any of these stories resolve.
My own theory is that the Emperor's threat will grow (or rather, more people will become aware that he intends to literally suck all life out of the Galaxy and become a lonely god) until eventually we'll all defeat him together in expansion content in a couple of years. Then a new Sith Empire will form, and some sort of detente will be reached. I honestly do not expect either side to "win" during the chronology of the game, because that removes much of the underlying tension.