The main objective of this paper is to introduce a new approach for constructing reversible and reversible DNA codes over the finite fields $F_{4^{2m}}$, $m\ge 1$ from any given polynomials by utilizing $T^n$-set codes. Notably, these polynomials are not required to be self-reciprocal divisors of $x^n-1$. In addition to this method, we provide some results that demonstrate how to generate reversible and reversible DNA codes from any $[n,k,d]$-cyclic codes over $F_{4^{2m}}$ by applying $T^n$-set codes. Moreover, this approach allows us to determine a lower bound for the distance before completing the entire calculation process. We also construct a correspondence table between DNA sequences made up of $20$-bases and all $256$ elements of the linear code $⟨T_g^5\mid T_{(g^{\ast }{)}^{\circ 4}}^5⟩$ over $F_{16}$, where $g=w^{13}+x+w^2{x}^2+w^3{x}^3$ is a polynomial in $x$ over $F_{16}$ and $(g^{\ast }{)}^{\circ 4}$ represents the Hadamard $4^{th}$-power of the reciprocal polynomial $g^{\ast }$ to demonstrate the fact that non-reversible codes may correspond to reversible DNA codes. Additionally, we provide a detailed table presenting some outcomes related to $l$-MDS codes, self-orthogonal codes, and their associated parameters.