  The proof of Witten's finiteness conjecture established that the Kauffman
bracket skein modules of closed $3$-manifolds are finitely generated over
$\mathbb Q(A)$. In this paper, we develop a novel method for computing these
skein modules.
  We show that if the skein module $S(M,\mathbb Q[A^{\pm 1}])$ of $M$ is tame
(e.g. finitely generated over $\mathbb Q[A^{\pm 1}]$), and the $SL(2,\mathbb
  C)$-character variety is reduced, then the dimension $\dim_{\mathbb Q(A)}\,
S(M, \mathbb Q(A))$ is the number of closed points in this character variety.
This, in particular, verifies a conjecture of Gunningham, Jordan, and Safronov,
that relates the dimension $\dim_{\mathbb Q(A)}\, S(M, \mathbb Q(A))$ to the
Abouzaid-Manolescu $SL(2,\mathbb C)$-Floer theoretic invariants, for large
families of 3-manifolds.
  We also prove a criterion for reduceness of character varieties of closed
$3$-manifolds and use it to compute the skein modules of Dehn fillings of
$(2,2n+1)$-torus knots and of the figure-eight knot. The later family gives the
first instance of computations of skein modules for closed hyperbolic
3-manifolds.
  We also prove that the skein modules of rational homology spheres have
dimension at least $1$ over $\mathbb Q(A)$.
