In particular, let $\mathfrak g$ be the semisimple Lie algebra of type $A_{2}$ et let $\alpha,\beta$ be its simple roots.

How can the multiplicity of weight $-2\alpha -3\beta$ be calculated in the Verma module $M(\alpha+2\beta)$?

As noted in the Wikipedia article on Verma modules, their definition relies on a stack of relatively dense notation. In particular, the following is the definition in my reference: $M\left(\lambda\right)=\mathcal{U}\left(\widetilde{\mathfrak{g}}\right)/J\left(\lambda\right)$, where $$ J\left(\lambda\right)=\mathcal{U}\left(\widetilde{\mathfrak{g}}\right)\hat{n}^{+}+\sum_{h\in\widetilde{\mathfrak{h}}^{*}}\mathcal{U}\left(\widetilde{\mathfrak{g}}\right)\left(\lambda-\lambda\left(h\right)\right)\subset\mathcal{U}\left(\widetilde{\mathfrak{g}}\right) $$

On the other hand, I have searched for useful Verma module formalisms and example solutions, and it seems that such information is scarce.