Let $X$ be a topological space with open cover $\mathfrak U$ and abelian sheaf $\mathcal F$. We know that if $\mathcal F$ is flasque, then higher Čech cohomology with respect to the covering $\mathfrak U$ vanishes, i.e. $\check{H}^p(\mathfrak U, \mathcal F) = 0$ for $p > 0$.
I am wondering if the same is true under different conditions. If $\mathcal F$ is merely acyclic in sheaf cohomology, I do not think that in general also higher Čech cohomology vanishes.
However is the above true, if $\mathcal F$ is fine, and $X$ is paracompact? I think I remember it to be true on smooth paracompact manifolds, but I am not sure if something changes in this 'general' case.
Thanks a lot in advance!