  In this paper, we prove new results on the geometry of the cuspidal
eigenvariety for $\mathrm{GL}_{2n}$ over a totally real number field $F$ at
classical points admitting Shalika models. We also construct $p$-adic
$L$-functions over the eigenvariety around these points. Our proofs proceed in
the opposite direction to established methods: rather than using the geometry
of eigenvarieties to deduce results about $p$-adic $L$-functions, we instead
show that -- via evaluation maps on parahoric overconvergent cohomology groups
-- non-vanishing of a $p$-adic $L$-function implies smoothness of the
eigenvariety at such points.
  More precisely, we attach a $p$-adic $L$-function to a non-critical
refinement $\tilde\pi$ of a regular algebraic cuspidal automorphic
representation $\pi$ of $\mathrm{GL}_{2n}/F$ which is spherical at $p$ and
admits a Shalika model. This gives the first construction of $p$-adic
$L$-functions in this generality beyond the $p$-ordinary setting. Further, when
$\pi$ has regular weight and the corresponding $p$-adic Galois representation
is irreducible, we show that the parabolic eigenvariety for
$\mathrm{GL}_{2n}/F$ is \'etale at $\tilde\pi$ over an
$([F:\mathbb{Q}]+1)$-dimensional weight space and contains a dense set of
classical points admitting Shalika models. Finally, under a hypothesis on the
local Shalika models at bad places which is empty for $\pi$ of level 1, we
construct a $p$-adic $L$-function for the family.
