"Interesting" categories whose internal logic is a dependent-linear type theory
Take any small symmetric monoidal category $V$.
Then the category of $V$-valued presheaves will (a) have closed monoidal structure (via Day convolution), and (b) have enough stuff (inherited from $\mathrm{Set}$) to interpret dependent types.  This gives you enough structure to interpret something like our LNL calculus pretty easily, because there is a nice adjunction between $\mathrm{Set}$ and this category given by the constant functor ($F(X) = V \mapsto X$) and the global elements functor ($G(A) = \Pi X : Obj(V). A_X$).
Matthijs Vakar wrote a nice paper, A Categorical Semantics for Linear Logical Frameworks, which describes the categorical models of this style of calculus in detail.