This work introduces a novel approach to constructing DNA codes from linear codes over a non-chain extension of $\mathbb{Z}_4$. We study $(\text{\textbaro},\mathfrak{d}, γ)$-constacyclic codes over the ring $\mathfrak{R}=\mathbb{Z}_4+ω\mathbb{Z}_4, ω^2=ω,$ with an $\mathfrak{R}$-automorphism $\text{\textbaro}$ and a $\text{\textbaro}$-derivation $\mathfrak{d}$ over $\mathfrak{R}.$ Further, we determine the generators of the $(\text{\textbaro},\mathfrak{d}, γ)$-constacyclic codes over the ring $\mathfrak{R}$ of any arbitrary length and establish the reverse constraint for these codes. Besides the necessary and sufficient criterion to derive reverse-complement codes, we present a construction to obtain DNA codes from these reversible codes. Moreover, we use another construction on the $(\text{\textbaro},\mathfrak{d},γ)$-constacyclic codes to generate additional optimal and new classical codes. Finally, we provide several examples of $(\text{\textbaro},\mathfrak{d}, γ)$ constacyclic codes and construct DNA codes from established results. The parameters of these linear codes over $\mathbb{Z}_4$ are better and optimal according to the codes available at \cite{z4codes}.