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Hans, I of course agree with Yemon Choi. If you find any true errors or confusions, please point those out in a comment, and I am happy to change my answer accordingly. Also, if my approach (essentially) answers your problem, why don't you accept it as 'answer'?
Are $\mu$ and $\sigma$ locally Lipschitz and in particular continuous? In this case, further assuming you have uniqueness of the martingale solution, Kallenberg Thm 21.11 gives the Feller property.
I think it is a reasonable question. To clarify, do you want a strict upper bound or instead the asymptotic behaviour as $n \rightarrow \infty$? A normal approximation of the Binomial random variable quickly gives you $E((X-n/2)^r) \approx (\sqrt{n}/2)^r (r-1)!!$, where $(r-1)!! = (r-1)(r-3) ... 1$.
