This paper presents an investigation of the secondary saturation characteristics of a HfTe₂ saturable absorber. Pulse energies of 5.85 mJ and 7.4 mJ were demonstrated with a high-order Hermite-Gaussian (HG) laser and a vortex laser, respectively, using Alexandrite as the gain medium. To the best of our knowledge, these are the highest pulse energies directly generated with HG and vortex lasers. To broaden the applications of high-energy pulsed HG and vortex lasers, wavelength tuning over a range of 40 nm was achieved using an etalon.