I've seen no evidence of it in canon, and the level of stability depicted in The Promise
would probably rule it out, but I still like the idea of some elements of the Fire Nation military refusing to leave the Earth Kingdom and setting up as self-sufficient warlords. They could last for decades on a continent the size of the Earth Kingdom, depending on how much of a nuisance they make of themselves.
And, I mean, there have to be a bunch of Hei Bai's out there, mad about their forests being burned down and taking it out on every Jim, Bob, and Sally who happens by.
But mostly I figure Aang spent most of his adult life as a nominal diplomat, having to be part of all international negotiations even if he has nothing to contribute. It would be part of re-establishing the authority of the Avatar (not very successfully, given how Korra is usually treated) and also a reminder to everyone to behave because he's keeping an eye on things.
"Avatar stuff" could have also been code for goofing off with his friends while the wife stayed behind and minded the more troublesome kids.
"Hey, Zuko, let's do Spring Break on Kyoshi Island! What? No, I'll just tell Katara it's 'Avatar stuff.' I'll take Tenzin as our alibi and leave him meditating in the hotel the whole time! ...what do you mean, Mai is too paranoid for that to work? What kind of a woman did you marry? ...oh, right, 'used to be Azula's best friend.' I keep forgetting that. Bummer, dude."