= Solution
The first assertion is false with the standard meaning of <linear equivalence of divisors>. For example, two distinct lines $D_1,D_2\subset\mathbb{CP}^2$ are smooth and linearly equivalent but intersect. Distinct fibers of a holomorphic map to $\mathbb{CP}^1$ are disjoint, so no map can have these lines as the fibers over $0$ and $\infty$. The ratio of defining sections gives only a meromorphic map, with an indeterminacy point at $D_1\cap D_2$. The assertion becomes true if the two sections have no common zero.
The requested cohomological conclusion is nevertheless valid. Linearly equivalent divisors define isomorphic holomorphic line bundles, and their Poincaré-dual Dolbeault classes both equal the <First Chern class> of that bundle. Hence $[D_1]=[D_2]$ in $H^{1,1}_{\bar\partial}(X)$.
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