I shot that combination for quite a while before moving to a D800.
There can be all sorts of reasons for your sharpness problem, including possibly a need to fine-tune your AF for that lens. But before you do that, some thoughts:
1. 500mm on that lens is NEVER going to produce the sharpest images. I found that zooming out even just to 450mm, or 400, helped--even when it meant cropping the image a bit. Of course, too much cropping can be a problem itself, when it comes to image sharpness. Experiment though with different focal lengths--are you having the problem with ANY shot over a certain focal length?
2. 1/125 isn't very fast, even for perched birds. I generally shoot perched birds at 1/500 and birds in flight (or otherwise moving) at a minimum of 1/1000. If it's really, really cloudy/rainy, I might chance going lower than those numbers, because the D7000 is really not that great in low light, and the Sigma 150-500 isn't either, so the combo really tends to work best in nice bright conditions.
3. You say you've tried different "modes" of shooting, but I'm not sure what you mean by that--like manual vs. aperture priority vs. shutter priority? Or different settings in manual?
Have you tried different AF settings? Different people have different preferences, but I tend to shoot birds on AF-S with a single focus point, aimed right on the eye.
4. Do some more experimenting. That first shot (which isn't bad at all), was f/11, 1/125, ISO 200, 500mm.
So take a similar shot at, say, f/8, 1/500, 400mm and set your camera to auto-ISO. Adjust one setting at a time--use f/8, f/6.3, etc. and take a few shots. THEN, set it back to f/8 and do a few shots with different shutter speeds, and then finally at different focal lengths. Now sit down and compare. If they are still ALL showing sharpness problems, it might well be time to fine-tune the AF.