Let $A_1$, $A_2$, $P$ be CPOs and let $\psi: A_1 \times A_2 \to P $ be a map, then $\psi$ is continuous $\iff$ it is so in each variable separately

I think you can justify the steps (if I didn't get wrong...)
Anyway, I'll start in a point you have already reached.

\begin{align} \psi \left(\bigvee D \right) &= \psi_{d_2}(d_1) = \psi_{d_2} \left( \bigvee D_1 \right)\\ &= \bigvee \psi_{d_2}(D_1) = \bigvee \{ \psi_{d_2}(x) : x \in D_1 \}\\ &= \bigvee \{ \psi(x, d_2) : x \in D_1 \}\\ &= \bigvee \left\{ \psi^{x} \left( \bigvee D_2 \right) : x \in D_1 \right\}\\ &= \bigvee \left\{ \bigvee \left\{ \psi^x(y) : y \in D_2 \right\} : x \in D_1 \right\}\\ &= \bigvee \{ \psi(x,y) : x \in D_1, y \in D_2 \}\\ &= \bigvee \psi(D). \end{align}