The initial value problem in Cauchy-type under the $(k,\psi )$-Caputo proportional fractional operators was our focus in this paper. An extended Gronwall inequality and its properties were analyzed. The existence and uniqueness results were proven utilizing the fixed point theory of Banach's and Leray-Schauder's types. The qualitative analysis included results for Ulam-Mittag-Leffler stability, which was also investigated. Using a decomposition principle, a novel numerical technique was presented for the $(k,\psi )$-Caputo proportional fractional derivative operator. Finally, theoretical results were supported with numerical examples to demonstrate their practical application, especially to blood alcohol level problems.