Hi Matt,
Two things spring to mind. One is that if you are zoomed in when focusing on the fast moving animals, the widest aperture you can achieve is f/6.3 on the 18-270mm. If its not a bright day, the camera's auto focus system may hunt for a target, which could explain the blurred images (if they are simply just out of focus).
The other is as you mention, shutter speed. The way around this is raising your ISO up to compensate. As the D7000 handles noise very well at higher ISO values, you should be able to boost the sensitivity up without compromising image quality to any significant degree. This may help you achieve a better shutter speed with the Tamron.
It might be worth experimenting with the D7000 at various ISO levels to see where you think the cut-off is for acceptable noise levels. Bear in mind that you can remove it to a large degree in post processing. This will help you discover the limits you can go to with the camera before image quality becomes affected.
Also, its worth shooting in RAW mode. If you opt for a high shutter speed in either S or M modes, and the resulting image is dark, you can probably recover virtually all lost detail in post processing due to the EV range of the D7000.
Ideally though, any fast paced action photography is shot using a lens that is designed for the task. Nature photography of wild birds is often done with 300mm+ F/4 prime lenses. For flexiblity, the 70-200 f/2.8 VR / VRII Nikkors + teleconverters, are often the lenses of choice. Fast focusing and good sharpness even wide open. Unfortunately, all lenses of this calibre carry a heft price tag. Unless you can afford the £1000-£1500 for a 70-200mm f/2.8 VR, you're limited to raising the ISO to achieve the stops of light that an f/2.8 lens would give you.
Hope that helps