It's certainly far from impossible, though.
The first thing you have to do when switching from tanking or healing to DPS or vice versa is abandon how you approached the job. You're not doing that job anymore. When I first went DPS in Firelands, it took me two weeks to get myself to stop trying to intercept mobs running for the healers or other DPSers and getting myself killed. That was because no one was healing me -- not because they didn't care, but because they had no idea I was about to get aggro on Firelands trash. Why would they? I wasn't a tank. It's not that they didn't appreciate it; it's that they had no way of anticipating I was going to do it.
Likewise, as a DPSer, you have a rotation or priority system you need to execute in order to get your DPS to where it needs to be. While specific fights control how much DPS you can actually do (and at times, you'll be told to control your DPS in order to not kill something), it's different from tanking (which is half holding threat via active use of abilities and half juggling incoming damage so you don't explode) and far less reactive than healing. Healers don't just run through their rotations -- they react to incoming damage and prepare for big spikes or lots of raid damage. Approaching one of the roles with the reflexes and habits of the others is something that will take you time to unlearn.
Some players are natural at this. Others have a longer breaking-in period. However, there are ways that you can shorten your adjustment time if you're having issues going from one role to another.
- Change your ability layout. Don't keep the same abilities on the same keybinds if there's any cross-use between roles. If you use similar abilities between roles with different names, make sure they're on different keys. Make it so that you actually have to stop and reorient yourself when you go from healing to tanking. Note that I am not suggesting you make it actively harder for you to find abilities you need, just make it different enough from your main spec or role that it requires you to think about it.
- Go out and practice. I'm not just saying hit a dummy (which would be useless for a healer, anyway); I'm suggesting that you run some Battlegrounds, hit a few heroic dungeons (or normal dungeons, if you're not comfortable with heroics or geared enough for them yet), and in general, use that new spec for its intended role. Once you're comfortable, hit up some Raid Finder runs and get a few stress tests in. It's not a bad idea to try and keep your gear and specs current by switching what you do in PUGs and Raid Finder as frequently as you can still enjoy. You don't even have to be intending to actually switch what your main spec is, just keeping yourself limber mentally by trying out the other role for a while.
- Learn from others. Even if they're not your class, if they're in your role, they probably have a few things to teach you. I'm not saying go ask everyone, but if your guild has a few veteran tanks or healers, why not see how they approach the role? Do they use a lot of macros? What addons do they use? How do they handle specific fight mechanics? Other players can be a resource -- you should be making use of it.
- Forget how you performed your old role. Especially if someone else is now doing it, don't spend even a second watching and saying, "That's not how I did it," because you're not doing it. Your attention should be on what you're supposed to be doing. Now, it's possible that you're right and that the new person is doing it wrong. Worry about that once you've mastered what you're supposed to be doing, not before.
- Pay attention to your mistakes. Yes, acknowledge them in raid or party, but also, really examine them. What did you do wrong? Why did you do it? Often, when you have a few seconds to dissect your mistake, it will end up breaking some misconception you had or habit you didn't even notice. Despite the way people can overreact to mistakes in PUGs and the Raid Finder, you're not a bad player for making one. You're a bad player if you refuse to learn from it. And when someone else points out a mistake, as long as they're not being a raging jerk about it, consider what they're telling you. Can you move in closer and pick up the adds sooner? Can you hold attacks for a few seconds? If the advice is warranted and good, make use of it. If not, you've lost nothing by considering it before rejecting it.
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