You have a good body. You want more reach... you don't need another body, you need another lens. THe 50-500 Sigma will also AF, wheras your 70-200 and 70-300 may struggle once teleconverted.
I agree - the 40D is a great camera (I still use mine), and quality reach seems to be your main shortcoming. But I'd recommend staying away from the Sigma 50-500. A non stabilised small aperture telephoto lens can be a bit of a handful. I guess you'll be using it for much more than just birds in flight? (BIF shots can only easily be pulled off with the new mode 3 IS)
I'd recommend pj1974's advice, sell up your two tele zooms, and get a 70-300L. As far as I know, the Sigma 50-500 is meant to be a short 500, so the difference in framing isn't as much as you'd expect between the two.
Even though the 70-300L isn't designed to work with TC's (it will at a pinch), the superior optics of the 70-300L should give you better quality if you crop to get the same narrow angle of view:
This comparison isn't a 100% match of what you could do as the 70-300 is on an 18MP crop (not a crop of a 10MP crop as I'm recommending), and the 50-500 is on FF (not an uncropped crop ), but the crop factor is roughly equal to what you could do in post. The 300 is cropped to a field of view of 480mm here, and it looks sharper than the 500 (in reality, just a 450-470mm) natively.