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First off, you obviously have to assume something about $-D$ not being effective, because otherwise you could take a negative ample class.

The cone of curves of a K3 surface is pretty well described in this paper. And there is a newer version of it that works in positive characteristic as well here.

Here is what you get out of this:

1. It is possible that there are no $(-2)$-curves on a K3 surface, but in this case for every divisor with $D^2\geq 0$ either $D$ or $-D$ is both nef and effective.
2. If the Picard number is $2$, it is possible that there is only one $(-2)$-curve. In this case there are actually (effective) curves with non-negative and even positive self-intersection which are not nef. However, they are negative on the sole $(-2)$-curve. (I think I will leave this for the reader for now).
3. In all other cases the $(-2)$-curves generate a cone which is dense in the cone of curves, so any divisor that is non-negative on the $(-2)$-curves is non-negative on every effective curve.

So, it actually looks like that what you want is true.