I'd get rid of all the dead material for sure. Yes, rooting hormone would help but over watering with cacti is a big deal as you've learned. Use a sterilized potting soil do not use garden soil. Make sure your saucer is always dry after watering! (get rid of the saucer and get the bottom of the pot off the surface so there is air between the bottom of the pot and its surface...or saucer!
Clay pots do best for cacti and sedums as they 'breathe'...these are very shallow rooted and if I were you I'd get a much shallower pot...wider than high for sure. Be careful about watering. Think of a quick spring rain in the desert...drainage is everything. Make sure you don't have any rocks or gravel beneath the soil as this actually creates a perched water table and the soil will become saturated before it drains out of the pot.
Don't water again until the soil is extremely dry but before the plant's skin starts getting dusky and puckered! Minimal fertilization, use an extended release fertilizer such as Osmocote and no more than every 4 months or what the label informs. Use bottled water, NOT TAP. Unless of course you have a well! Even then try to find bottled water without fluoride...as you should for your own needs. Fluoride is an idiotic additive supposedly for our teeth but NOT!! It is a toxic chemical for us humans and plants...crazy, huh!