Generalizing Frassek et al. (Adv. Math. 401, 108283 (2022). https://doi.org/10.1016/j.aim.2022.108283), we construct a family of SO(2r), Sp(2r),  $SO(2r\!+\!1)$ rational Lax matrices $T_D(z)$, polynomial in the spectral parameter z, parametrized by $\Lambda ^+$-valued divisors D on ${\mathbb {P}}^1$. To this end, we provide the RTT realization of the antidominantly shifted extended Drinfeld Yangians of ${\mathfrak {g}}=\mathfrak {so}_{2r}, \mathfrak {sp}_{2r}, \mathfrak {so}_{2r+1}$, and of their coproduct homomorphisms.