  We consider quantum integrable models associated with $\mathfrak{so}_3$
algebra. We describe Bethe vectors of these models in terms of the current
generators of the $\mathcal{D}Y(\mathfrak{so}_3)$ algebra. To implement this
approach we use isomorphism between $R$-matrix and Drinfeld current
realizations of the Yangians and their doubles for classical types $B$, $C$,
and $D$ series algebras. Using these results we derive the actions of the
monodromy matrix elements on off-shell Bethe vectors. We show that these action
formulas lead to recursions for off-shell Bethe vectors and Bethe equations for
on-shell Bethe vectors. The action formulas can also be used for calculating
the scalar products in the models associated with $\mathfrak{so}_3$ algebra.
