This study presents new closed-form estimators for the Dirichlet and the Multivariate Gamma distribution families, whose maximum likelihood estimator cannot be explicitly derived. The methodology builds upon the score-adjusted estimators for the Beta and Gamma distributions, extending their applicability to the Dirichlet and Multivariate Gamma distributions. Expressions for the asymptotic variance-covariance matrices are provided, demonstrating the superior performance of score-adjusted estimators over the traditional moment ones. Leveraging well-established connections between Dirichlet and Multivariate Gamma distributions, a novel class of estimators for the latter is introduced, referred to as "Dirichlet-based moment-type estimators". The general asymptotic variance-covariance matrix form for this estimator class is derived. To facilitate the application of these innovative estimators, an R package called estimators is developed and made publicly available.