  In this paper we introduce the shifted twisted Yangian of type {\sf AI},
following the work of Lu--Wang--Zhang, and we study their semiclassical limits,
a class of Poisson algebras. We demonstrate that they coincide with the Dirac
reductions of the semiclassical shifted Yangian for $\gl_n$. We deduce that
these shifted twisted Yangians admit truncations which are isomorphic to
Slodowy slices for many non-rectangular nilpotent elements in types {\sf B},
{\sf C}, {\sf D}. As a direct consequence we obtain parabolic presentations of
the semiclassical shifted twisted Yangian, analogous to those introduced by
Brundan--Kleshchev for the Yangian of type {\sf A}. Finally we give Poisson
presentations of Slodowy slices for all even nilpotent elements in types {\sf
B}, {\sf C}, {\sf D}, generalising the recent work of the second author.
