Well I play both, and until MW3, was fairly effective at both. (I've only put about 3 hours into MW3, but just now managing to even be an asset to my team, other than letting them know where I was killed from, lol).
I tend to use the same shooting mechanics as you in CoD. But I not in BF3. Generally you are not close quarters, so I tend to wait, get my shot lined, then fire. You've got to know when it's worth taking the shot and when it's not.
In close quarters, I tend to just stay at the hip.
Mainly, the difference should not be in firing as much, but in how you move. Screw fair fights, be patient and sneaky, get up behind enemies or engage the enemies firing on your teammates.
I have not noticed the issue you have been having, but I think maybe taking better advantage of cover and map navigation could help. You sound like you have a good handle on aim and accuracy, so, you're probably just engaging people with weapons of higher damage or they have cover when you do not. Or possibly engaging multiple enemies at once by yourself.
Instead of worrying about the gun mechanics, you could try navigating the map differently and see if that helps.
Good luck