Can I still save this mushy-leaved succulent?

Take it out of the wet soil and check the stem and remaining roots closely. Remove any roots that are black, slimy, or clearly rotting with clean scissors, then let the plant and roots dry before potting it in a small pot with a drainage hole and a dry, gritty mix. Keep it in bright indirect light while it settles. Give damaged roots several days to heal before watering, and don't water again until the mix has dried completely.

A firm stem is a hopeful sign, but it can't tell you for sure how much damage is below the soil. Mushy, dropping leaves can fit with excess moisture, though damaged leaves won't firm back up. Focus on whether the stem stays firm and whether the rot has reached it. If the stem turns soft or dark, cut above that area into firm, clean tissue and let the cut dry before trying to root it.

I'd avoid giving it a big drink right after repotting. If the roots are stressed, more moisture can make things worse. Use a fast-draining mix, a pot that isn't oversized, and patience. Once it's settled and the soil is fully dry, water thoroughly and let excess drain away. If it keeps declining, a healthy firm piece of stem may be worth trying as a backup cutting.