Get used to maxing your camera distance and SPINNING it around constantly. This has helped me so much. Especially if you are a ranged dps class you should learn to do you job without having to constantly look at your target. When doing 5 mans practice and see if you can spin your camera to look behind you for most of the fight, just glancing back to check positions and such.
Also try to move using your "q" and "e" keys more. This keeps you facing your target while repositioning. This helps on fights like VR - move away from the orb coming at you, but make sure you don't move right into another one. It's not real life, you arn't going to trip over something if you don't look where you are going!
Think about the advantages this would give you in say... i don't know, the Vashj fight. If you have your camera maxed you will see everything that spawns in your area and you greatly reduce the risk you will miss a core. Spinning you camera means you can keep tabs on where the strider is so you don't miss your dots and where you healer is in case you need to toss the core to them.
The more you know about the fights and your job in them, the less you brain has to focus on that. When you can do a fight without even thinking about it, it totally frees you up to watch what is going on and adapt.
Also, if you are a clicker, try to fix this. You can't spin your camera if you are using your mouse to cast spells. It's not a hard rule, but try, as a rule of thumb, to use your mouse hand to move and your keyboard hand to cast all spells. There are some spells you only use once a fight or so, I click them, but your core spells shouldn't be cast via mouse. Some players even move wasd to esdf to make 1-5 easier to reach instead of just 1-4. I never could get used to it and can easily reach up to 6 without moving my hand, so it's all about your play style. Just work to improve it - don't say "well, I'm just not as young/as much of a gamer/as good as so and so" and just be satisfied with your current style.
Improving your skill not only helps us all move forward and do better in the game - it really really does make it more enjoyable.