Good luck on that 'raying'...
I play with a priest, not a druid, so my tips might not be good for you, but here goes anyway.
The main problem is that warriors are just a superior class to use in 2v2. They do burst dps where we do sustained, they have better mobility and in this setting, there isn't enough 'damage' around to truly make their 10% increased damage from berserker stance a weakness, especially since any 2s with a rogue in it has to deal with their armor. Improved Hamstring and Mace stuns are just icing on the cake.
Our strategy so far has been to ignore the warrior and go for the healer. Getting on the warrior usually means just feeding him rage, and rogues are pretty damn ineffective at stopping a warrior anyway. Our stuns gives them back rage, their favorite teammates can all take care of crippling poison and Intercept mean that eventually, they'll simply get away.
Sadly, this is easily counter-able by any warrior that has a clue. Just abuse Intercept and Intervene, keeping me snared (with possible mace stuns) and going back to the priest. Or hell, just stay on the rogue. Eventually he'll run out of way to avoid damage and then he'll get stunned/snared to oblivion.
However, at our current rating (My new priest and I hit 1860 yesterday) it works very well, because a lot of warriors are pretty bad: Even at 2ks rating in previous seasons, we'd often bump into warriors that were keyboard turners. Tunnel vision is also extremely frequent. Basically, you've got to play your A-game and hope the warrior team screws up, because if they also play their A-game, they will win.
Warrior + Priest: The easiest one. Between Mind-numbing and Wound Poison and kick/gouge, I can nullify a priest's healing even better then a Warrior can lock down my priest's. Aggressive use of Vanish for Garotte is key here as well as the proper timing of your kidney shot.
Warrior + Shaman: Once again, it all depends on how smart the warrior is. And how lucky he is. On top of mace proc and imp hamstring procs, you now got Windfury totem kicking in. RNG 4tw. Getting a early dispel to take care of earthshield helps a lot. If your priest can kite the warrior outside of WF totem range, even better. As the Shaman drops down in help more, you want your priest to come back in to dispel ES again, so you can use a Vanish/CS/KS (try to get EA up) and finish him off.
Warrior + Druid: On this one, a lot of the pressure to not ***** up is actually on the druid. My priest is human, so he'll usually pop perception when the warrior engage him and try to dot the druid out of stealth (They quite often prowl near their warrior)... getting the opener on the druid is huge, but quite often I'll have to break stealth and get on the warrior to save my priest.
A lot of druid don't seem to realize they have bear form and that between their armor, resilience and Natural Perfection, they have almost 85% physical damage mitigation in that form. Instead, they keep trying to run away by switching between Caster and Travel form. So stay on them. Use shadowstep/vanish/sprint and don't be afraid to prep. Try and KS when in travel form if at all possible and have your priest friend toss in some damage (even if it's just sw:pain). Even the dimmest warrior will usually intercept you/intervene his druid once to try and protect him... but that's okay, if he only does it once or twice. The bad one do it once or twice and then forget about you, seemingly thinking their druid buddy is fine now. The smart warriors simply never stop, and eventually you run out of tricks.
Those two class in 2v2 are so strong however, that even if the warrior isn't good, they can quite possibly beat you. They have tons of room for error and it truly doesn't take much time for a druid to heal himself back to full if you aren't constantly glued to him.
Warrior + Paladin: I'm starting to think this combo is actually better than druid/warrior. It's certainly rarer now, but it seems those guys are always good... which I guess makes sense, when you play a comp with an inherent weakness, you need to make up for it somehow. Our success rate against this comp has been the worse of all the Warrior + Healer so far.
The 'go for the healer' strat doesn't work to hot when the healer is in plate. Going to need to figure out something here.