I rolled my Shaman to 60 as Enhancement (Windfury while leveling is one of the great pleasures of Azeroth), but when I hit 60 way back when, I switched to Resto -- I like playing in groups, and being a healer makes sure you have groups whenever you want them. But when Burning Crusade came out, I still wanted to play instances, so I leveled from 60-70 as Resto.
How'd I do it?
First and foremost, you've got to get yourself a good set of DPS gear that capitalizes on the damage spells you can do. And when I say a "set," I mean a full set -- trinkets, gemmed gear, the works. Aim for spell damage (because you're only blowing through daily quests, you won't have to worry so much about Intelligence or Spirit, though those do help you kill faster in the right situations), and get as much of it as possible. Even if you can't nab DPS gear as a healer on the raids you're doing, a Karathon can net you a good amount of gear these days, and everyone's got so much of it that it's easy to pick up a DPS set even if that's not your spec, too. Badges are helpful as well -- use your healing powers to run a few Heroics, pick up badges aplenty, and get all that sweet badge gear Blizzard put in.
Other than that, there are a few good dailies that don't actually require you to do much casting -- bombing runs, the fishing dailies, and so on. Some collecting quests, as a priest, are pretty easy to do even with just fearing and running away with the loot. And if nothing else works, there's always the daily Heroics and BGs -- you already know being a healer comes in handy while running instances, and people like to see healing in the BGs as well.
In short, there's lots to do as a healer. It can be kind of fun to push your character to do something they're not specced for, but even if you're not up for that, +healing doesn't always have to be a disadvantage in daily quests.