We present a machine-learning approach for estimating galaxy cluster masses, trained using both Chandra and eROSITA mock X-ray observations of 2041 clusters from the Magneticum simulations. We train a random forest (RF) regressor, an ensemble learning method based on decision tree regression, to predict cluster masses using an input feature set. The feature set uses core-excised X-ray luminosity and a variety of morphological parameters, including surface brightness concentration, smoothness, asymmetry, power ratios, and ellipticity. The regressor is cross-validated and calibrated on a training sample of 1615 clusters (80% of sample), and then results are reported as applied to a test sample of 426 clusters (20% of sample). This procedure is performed for two different mock observation series in an effort to bracket the potential enhancement in mass predictions that can be made possible by including dynamical state information. The first series is computed from idealized Chandra-like mock cluster observations, with high spatial resolution, long exposure time (1 Ms), and the absence of background. The second series is computed from realistic-condition eROSITA mocks with lower spatial resolution, short exposures (2 ks), instrument effects, and background photons modeled. We report a 20% reduction in the mass estimation scatter when either series is used in our RF model compared to a standard regression model that only employs core-excised luminosity. The morphological parameters that hold the highest feature importance are smoothness, asymmetry, and surface brightness concentration. Hence these parameters, which encode the dynamical state of the cluster, can be used to make more accurate predictions of cluster masses in upcoming surveys, offering a crucial step forward for cosmological analyses.