When I got my 400/5.6 one of the first things I did was a test between it and my Sigma 150-500. I printed out a lens test chart from the internet, shot it with both lenses from the same distance, and made 8x10 prints of a very small part of the frame from each shot. The shots from the Canon had to be cropped tighter and blown up further for the prints from each lens to have the same image at the same size. The shots taken with the Canon lens were significantly sharper, even though cropped tighter and blown up more. In other words, at least with my versions of these lenses, I'm getting sharper images with the shorter lens cropping tighter.
And the focusing speed and accuracy of the Canon lens is far superior.
I think in your situation your best bet is to stick with the 100-400, and learn to use it within its reach limitations. I don't think you'll find anything better (and likely, not as good) in the budget you've talked about. The next real step up is probably a Canon 500mm or 600mm, serious money.
A lot of people are really happy with the Sigma 150-500, and it really opened the door to bird photography for me when I got mine, but I've since met quite a few people who have, like me, moved from it to the Canon 400/5.6 or the 100-400.
This is a shot from last fall at Conowingo dam: