The musty-smelling soil is your main problem - it points to waterlogging and root rot, not primarily a fungus. Echeverias are succulents and really need very little water, but what's even more important is that the soil has to dry out completely between waterings. If the soil smells musty, the roots are probably already rotting, and that's what's causing those soft, brown spots you're seeing.
Take the plant out of the pot and check the roots. If they're dark, mushy, or smell musty, you need to cut away the rotted parts with a clean knife and repot the echeveria in fresh, well-draining soil (ideally special succulent soil or regular soil mixed with a lot of sand). Use a pot with drainage holes and don't water for the next two to three weeks - let the cut areas harden off first.
Your location could also be playing a role. Echeverias love bright light and need good air circulation; if your windowsill isn't particularly airy or is too dark, that makes the rot problem worse. After repotting, you should plan for longer dry periods and only water again when the soil is really dry - two to three weeks is a good rhythm, but in winter it can be even longer.