Fractional delay differential equations constitute a powerful mathematical framework for modeling complex dynamical phenomena exhibiting memory and delay effects. In this study, we investigate a class of fractional delay differential equations incorporating Caputo and Riemann-Liouville fractional derivatives with a delay term. Unlike previous approaches, we establish the existence and uniqueness of the analytical solution under relaxed Lipschitz conditions on the nonlinear terms, without requiring contraction assumptions. Utilizing Picard iteration techniques, we demonstrate convergence of the numerical method under these Lipschitz conditions, thereby broadening the applicability of our model to a wider range of real-world scenarios. Additionally, numerical tests are conducted to validate the effectiveness and accuracy of the proposed method, further highlighting its utility in practical applications. Our findings offer new insights into the modeling and analysis of complex dynamical systems, with implications for various scientific and engineering disciplines.