Equations for the degree 5 case are in arxiv.org/abs/1812.10715. Equations in a product of lines are in arxiv.org/abs/1803.02984 section 3

This is my point. After the work of Manetti and Catanese and others we have simply connected topological manifolds with two complex structures that are not deformation equivalent. I would expect that their examples may be adapted to characteristic p to give a negative answer to your question, but I'm not an expert in finite characteristic issues.

Why should that be true?

I think I misread your question. I interpreted locally constant as locally trivial, I mean invertible. Sorry

I'm not sure what you mean by $Y$, but I think here you are wrong, this factorization is biregular. A bidouble cover $X \rightarrow Z$ can be factored as a composition of two double covers in three different way, say $X \rightarrow Y_i \rightarrow Z$: if $L_1,L_2,L_3,D_1,D_2,D_3$ are the building data of $X \rightarrow Z$, then $f_i \colon Y_i \rightarrow Z$ is the double cover branched on $D_j+D_k$ (here $\{i,j,k\}=\{1,2,3\}$) whereas the double cover $X \rightarrow Y_i$ is branched on the pull-back of $D_i$.