  We generalize some known results for limit groups over free groups and
residually free groups to limit groups over Droms RAAGs and residually Droms
RAAGs, respectively. We show that limit groups over Droms RAAGs are
free-by-(torsion-free nilpotent). We prove that if $S$ is a full subdirect
product of type $FP_s(\mathbb{Q})$ of limit groups over Droms RAAGs with
trivial center, then the projection of $S$ to the direct product of any $s$ of
the limit groups over Droms RAAGs has finite index. Moreover, we compute the
growth of homology groups and the volume gradients for limit groups over Droms
RAAGs in any dimension and for finitely presented residually Droms RAAGs of
type $FP_m$ in dimensions up to $m$. In particular, this gives the values of
the analytic $L^2$-Betti numbers of these groups in the respective dimensions.
