Killing a disc Priest with a Resto Shaman / Afflic Lock isn't that difficult, your Shaman just has to realize that there's no threat to his life and he can play 100% aggressively. He needs to purge everything the moment it's applied, and windsheer anything with a cast time while using Frost Shock as much as possible to help you keep him slowed. Basically a Disc Priest has his shield, renew, and prayer of mending that are instant casts.. these can all be purged (although getting PoM before it bounces when there's dots ticking is going to be extremely difficult.) Beyond that he has to cast to heal himself. Wind sheer is widely considered one of the most powerful abilities in the game, even by Shamans themselves.. having him interrupt cast time spells is key to getting the kill.
Your first job is getting him out in the opened with a good fear bomb which will allow you to get full dots ticking on him, then hitting him with Shadowflame every CD. I realize this will put you in range of a fear, but you will be dispelled almost instantly if your partner is doing his job. You should use curse of tongues whenever he isn't feared.. this will buy more time for those last second interrupts and buy your Shaman more time to clean off those shields, renews, PoMs before another is applied. You then need to focus on keeping your dots up, keeping haunt up, keeping him in LoS (with fears, shadowflames, etc) and saving your deathcoil for that moment when he's low but he has a shield around his cast bar.
It seems like your Shaman wasn't doing his job.. he needs to be purging anything he can and use Windsheer every time the Priest casts while using Frost Shock on CD and keeping an Earthbind down near the Priest to make sure the Priest never really moves fast. He also needs to make sure he's not ever sitting on top of the Priest so that when you get feared he can just dispel it instead of both of you eating a full duration fear and letting him get back to 100% hp. Whenever he can, he needs to be throwing as much damage out as possible but this is nowhere near as important as getting off purges (what's more important, finishing the last .5s of that lightning spell that will do 4k damage and just be absorbed or removing that shield that will absorb 25k damage? it's a no brainer.)