I would go for the A6300. Full-frame is all well and good, but with today's technology you have to ask yourself whether you really need it. The AF should be a bit faster on the A6300.
But for sports and fast-moving images, I wouldn't use either of them... I'd rather buy a Fuji X-T2 for the money... Some of the lenses there are much cheaper and the quality of a Fuji can easily keep up with Sony.