In the present article, linear positive operators connected with the adjoint Appell-Euler polynomials are proposed. Their moments are calculated and quantitative convergence estimates are provided using the $K$-functional and various moduli of smoothness such as first-order, exponential, and weighted moduli. Furthermore, Korovkin type theorems are established for different sets of test functions. Moreover, the associated modified operators are introduced which preserve constant functions and $e^{2 \mu x}, \ \mu >0$. Asymptotic formula and some analogous convergence results are established for these operators. Finally, we examine the convergence behaviour of both operators through numerical tables and graphical examples.
