  Crystal basis theory for the queer Lie superalgebra was developed by
Grantcharov et al. and it was shown that semistandard decomposition tableaux
admit the structure of crystals for the queer Lie superalgebra or simply
$\mathfrak{q}$-crystal structure. In this paper, we explore the
$\mathfrak{q}$-crystal structure of primed tableaux (semistandard marked
shifted tableaux) and that of signed unimodal factorizations of reduced words
of type $B$. We give the explicit odd Kashiwara operators on primed tableaux
and the forms of the highest and lowest weight vectors. We also give the
explicit algorithms for odd Kashiwara operators on signed unimodal
factorizations of reduced words of type $B$.
