I can't think of a campaign which is completely cooperative (unless it's the players against an umpire), but there are those where all the players are on the same side, but in competition with each other at the same time.
There is the Race to the Rhine campaign published in Battlegames magazines a few years back where players are Allied commanders moving westwards on parallel columns on a hex map. IIRC, they roll for chance of encounter each turn, and the enemy is played by another player/umpire. They are nominally on the same side, but do not interact as allies directly, and are in competition with each other, each wanting to be the first to cross the Rhine.
Then there is "Stonewall Jacksons Shenandoah Valley Campaign" by Realistic Modelling.
link
This has Union commanders trying to find and engage Confederate armies on a node-map of the Shenadoah Valley, at the same time trying to raise their political prestige back in Washington.
The Confederate army's movements are controlled by an algorithm and dice, so a player only knows the rebel army's last known location, and can march towards/away from it, but will not know if it is still there until they reach a node, or which Commander/Divisions/Brigades he will actually find there. When battle results, another Union player plays the Confederate army.
Players gain prestige for success in battle and playing politics, so battlefield success is not the only victory condition here.
I think a similar model may work for an 1813 campaign, with each Coalition commander trying to be the one to find and defeat the retreating (and occasionally counter-attacking) French army. I do recommend you get a copy of the rules for reference.
Then there is my own unpublished First Crusade campaign inspired by Thomas Asbridge's book. Players are the crusader commanders, with no clear leader and binding alliance amongst them. The eventual aim is to take Jerusalem (and all the towns along the way). I used the Crusader Rex map from Columbia Games. Players gain prestige for taking towns, and each town has a number of points. However, if they do not agree on how to split the points, no one gets any. This gives them the option of going their own way, or cooperating (as was the case back then). Armies are not fixed, but generated at each battle based on the current prestige points (reflecting the shifting loyalty of the followers), and the Saracen army is rolled for each time too, depending on the size of the crusader army that arrives at a town (reflecting their tendency to be complacent until faced with a definite threat) – sometimes the crusaders will find the town abandoned or surrendering without a fight. Again, a non-participating Crusader player plays the Saracen army when battle occurs. I am quite proud of these rules as I think they recreate the operational level decisions of First Crusade as described by Asbridge quite well.
This model is more suited to a medieval era.
Hope that helped.