  We define the Drinfeld generators for $Y_3^+$, the twisted Yangian associated
to the Lie algebra $\mathfrak{so}_3(\mathbb{C})$. This allows us to define
shifted twisted Yangians, which are certain subalgebras of $Y_3^+$. We show
that there are families of homomorphisms from the shifted twisted Yangians in
$Y_3^+$ to the universal enveloping algebras of various orthogonal and
symplectic Lie algebras, and we conjecture that the images of these
homomorphisms are isomorphic to various finite $W$-algebras.
