If the base is mushy, try checking the root to detect any rotting. If it is rotting (will give out a bad smell as well), you may have to re-plant the non-mushy stem again. I hope I'm right, but this is what I'd done. I have messed up with aloe growing in the past, now I've realized that reviving a dying plant when I have under-watered is easier than saving it when I've over-watered. Anyone with more experience out there, please correct me if I'm wrong about the re-planting bit.