Show that a functor which preserves colimits has a right adjoint

This is false in general without further hypotheses; see adjoint functor theorem. The reference to CWM, as you say, is only a reference for the uniqueness. The various adjoint functor theorems do imply this statement for a functor between Grothendieck topoi.

I don't know a reference for elementary topoi, but one for presheaf topoi can be found around pages 41-43 of MacLane-Moerdijk Sheaves in Geometry and Logic: they prove (Corollary 4) that a functor $A:\mathbb{C}\rightarrow\mathcal{E}$ where $\mathcal{E}$ is cocomplete and $\mathbb{C}$ is small extends along the Yoneda embedding $y$ to a unique (up to iso) $L_A:\widehat{\mathbb{C}}\rightarrow\mathcal{E}$ which preserves colimits, and such a functor has a right adjoint by construction (Theorem 2).

If now you consider some colimit preserving functor $F:\widehat{\mathbb{C}}\rightarrow\widehat{\mathbb{D}}$, the composition $A=Fy$ must extend in such a way: so it extends to $L_A=F$, but then $F$ must have a right adjoint.