Not a complete answer (for lack of time, will try to add later). Since $\min(a,b)=(a+b-|a-b|)/2$, your question is really about estimating $E|2X-n|$, or, equivalently, $E|X-n/2|$. For $1/n<p<1/2$, this should be roughly like the variance, or order of $\sqrt{np}$.