Are you shooting BIF (Birds in Flight) or Birds sitting on branch? Could you upload a few examples to your gallery and provide a link to them? What are you using to support (tripod and head) your 300mm f/2.8 and TC? A tripod with 32mm or larger leg diameter (Gitzo 3 Series) along with a Wimberley Sidekick or a Full Gimbal mount will allow unrestricted movement and the required stability.
What Shutter Speeds are you using?
For BIF Use: Continuous AF (AF-C) mode.
a1: AF-C Priority Selection = Release + focus or Focus When set to Release (default) the shutter will release as soon as the button is pressed regardless of focus. When set to Release + focus the FPS rate will slow down for improved focus on dark or low contrast subjects. When set to Focus the shutter will only release when the focus dot is dislpayed. This will increase your in-focus shots but may not allow you to capture the BIF.
a3: Dynamic AF Area = 9 points
Always use the minimum number of AF points required for any given subject.
a4: Focus Tracking with Lock-On = Normal or Long
When set to Off, when a branch or something else enters the frame between you and your subject the camera will imediatly refocus. The same will happen if the active AF point(s) lose track of the subject and the trees behind it has more contrast. Setting Normal or Long gives you more time to re-establish focus on your subject if you momentarily lose track.
For shooting static subjects use a Single AF point and either AF-C using the AF-On button or AF-S mode. Good Luck and Enjoy your Nikons!