The amount of blurriness will depend on the lens not the camera, and a good lens will get you a sharp image even on a D3300. I have used the Nikon 200-500 on a D3200, and although it can be hard to nail the focus with that big lens and that body, it can be done, and it can be plenty sharp. Some of it is a matter of practice, and finding the most suitable AF mode for what you're after.
When I got that lens Tokina had not come out with its newer 150-600 and the older one, though decent, was not as good and could not match the Nikon for vibration control. The newer one looks like more competition. I have not tried either version of the Sigma. The Nikon and Tamron were both available to try in real life, and I found the Nikon just very nice to use, so I took a deep breath and bought it. I like it a lot.
As for the distortion on the lens you have, I'd not worry about what the reviews say. Distortion will depend on what you're shooting. Duke Ellington famously said of music, "if it sounds good, it is good." I'd suggest the same for how a photograph looks. If you're taking bricks or clapboards, distortion and edge fuzziness are a big deal. If you're taking a bird on the wing, they're probably not. If you find the distortion or lack of sharpness bothers you then by all means look for something else, but make sure first that you're seeing real faults in the lens and not in other circumstances.