Both of my characters have carried on adventuring after the Main Quest.
With Relanna, I kicked around doing lots of other quests before getting around to finishing the MQ. RP-wise, I spun it as if Jauffre had said "thanks for the courier service" and sent her on her merry way after she delivered the Amulet of Kings. Many quests and a significant rise in rank in the Mages' Guild later, Jauffre called on her again when the siege of Kvatch began. After that I more or less plowed through the MQ, although I took some breaks where it seemed timely to explore and do other quests, including rebuilding Kvatch (yay mods). All told the Oblivion Crisis lasted well over a year in that game.
Afterwards it was easy for me to keep going; Relanna was a power-hungry mage and thought for sure she'd be serving at the new Emperor's right hand, at the very least. Instead she got a pat on the head and some mostly ceremonial armor. ...She was displeased.
The game took a darker turn after that, as she finished the Dark Brotherhood quests, took over the Mages' Guild, wandered into the Shivering Isles, and (thanks to more mods) eventually stole the High Chancellor's seat out from under Ocato.
Then I started thinking about retiring her.
My second character, Amarel, being a much nicer and more law-abiding sort, gave the Main Quest priority over almost everything else. If she came across a quest where someone direly needed help and didn't need her to march across half of Cyrodiil she'd stop and assist; otherwise, everything was post-poned. The Oblivion Crisis lasted a little over a month, in her case. Now that the MQ is done she's finally getting a chance to turn her attention towards those other quests and exploration. For her the end of the MQ was something of a relief on some level, since her life and time was finally her own again. And yes, helping the Fighters' Guild out with some terrifying rats
might be a little anti-climactic, but it's honest work, and she's glad to be finally pulling her weight in the guild.