  Results of Morse and Schilling show that the set of increasing factorizations
of reduced words for a permutation is naturally a crystal for the general
linear Lie algebra. Hiroshima has recently constructed two superalgebra
analogues of such crystals. Specifically, Hiroshima has shown that the sets of
increasing factorizations of involution words and fpf-involution words for a
self-inverse permutation are each crystals for the queer Lie superalgebra. In
this paper, we prove that these crystals are normal and identify their
connected components. To accomplish this, we study two insertion algorithms
that may be viewed as shifted analogues of the Edelman-Greene correspondence.
We prove that the connected components of Hiroshima's crystals are the subsets
of factorizations with the same insertion tableau for these algorithms, and
that passing to the recording tableau defines a crystal morphism. This confirms
a conjecture of Hiroshima. Our methods involve a detailed investigation of
certain analogues of the Little map, through which we extend several results of
Hamaker and Young.
