I can feel that my hand is unsteady - feels like the camera is shaking (can't be good!). I suppose a fast shutter speed should compensate? The lens is new - I think I need to start lifting a few weights and get used to it 
Lifting weight would give you staminia, which is good, but steadiness comes much more from knowing how to brace yourself. For static subjects, having a solid place to rest your elbows would help a lot. If your body is flexible enough to stay in squat position, all of a sudden you have a pair of knees to put your elbows on. Even if you just raise one knee in a kneeling position, you still have one place to put the elbow that supports the weight of the camera. And if you're on the ground in prone position, of course you can plant your elbows solidly on the ground. So low positions are good for stability. Plus they also give a natural look to birds on the ground or in water.
The only time you really want to keep the camera high is when your subject is high--on a branch or fence for example. In that case, it's difficult to support the elbows properly. A monopod would be a good idea here.
But for flight shots, you want to stand up and without support exactly because it allows you to freely point in whatever direction you want. Also move your left hand way out to the front of the lens will give you better control. Here's the ideal form as demonstrated by the (in)famous peacefrog:
http://www.pbase.com/dadas115/image/25905696Andy
