When shooting at 300mm, f5.6 handheld without IS or OS, it doesn't matter if it is a $300 lens or $1200 lens, the number of keeper in general won't be high dues to camera shake, IMHO. Fortunately there are a few things you can do to increase the %. The cheapest solution is find something you can brace on when shooting, such as wooden fence, big rock, cement block...etc to get more stability. You can add a bean bag as a support.
Or, use a tripod or monopod. Get a ballhead of you pan a lot. I shoot mostly birds with my Sigma 50-500mm. The lens is mounted on my tripod pretty much 99% of the time. What a huge difference it made compared to the photos taken before tripod.
Or, get a tele lens that comes with IS or OS. Then you don't have to worry about tripod or monopod.
The bottom line is, make the camera more stable when you press the shutter button. I have seen great results coming out from the Sigma 70-300mm when right technique is applied. Spend some time and learn the lens and you shall be rewarded. 