We construct a family of $G{L}_n$ rational and trigonometric Lax matrices $T_D(z)$ parametrized by ${\mathrm{\Lambda }}^{+}$-valued divisors D on ${\mathbb{P}}^1$. To this end, we study the shifted Drinfeld Yangians $Y_{\mu }({\mathfrak{gl}}_n)$ and quantum affine algebras $U_{{\mu }^{+},{\mu }^{-}}(L{\mathfrak{gl}}_n)$, which slightly generalize their ${\mathfrak{sl}}_n$-counterparts of [3,18]. Our key observation is that both algebras admit the RTT type realization when μ (resp. ${\mu }^{+}$ and ${\mu }^{-}$) are antidominant coweights. We prove that $T_D(z)$ are polynomial in z (up to a rational factor) and obtain explicit simple formulas for those linear in z. This generalizes the recent construction by the first two authors of linear rational Lax matrices [15] in both trigonometric and higher z-degree directions. Furthermore, we show that all $T_D(z)$ are normalized limits of those parametrized by D supported away from $\{\infty \}$ (in the rational case) or $\{0,\infty \}$ (in the trigonometric case). The RTT approach provides conceptual and elementary proofs for the construction of the coproduct homomorphisms on shifted Yangians and quantum affine algebras of ${\mathfrak{sl}}_n$, previously established in [14,18] via rather tedious computations. Finally, we establish a close relation between a certain collection of explicit linear Lax matrices and the well-known parabolic Gelfand-Tsetlin formulas.