Recently, Alanazi, Munagi, and Saikia employed the theory of modular forms to investigate the arithmetic properties of the function $\overline{R_{\ell,μ}}(n)$, which enumerates the overpartitions of $n$ where no part is divisible by either $\ell$ or $μ$, for various integer pairs $(\ell, μ)$. In this paper, we substantially extend several of their results and establish infinitely many families of new congruences. Our proofs are entirely elementary, relying solely on classical $q$-series manipulations and dissection formulas.