My central air smells like sulfur or rotten eggs. Why

You need to flush out the system as water must have gotten in the system hence the bad smell. Don't use a hose inside your home, but use something like a pitcher to gently clean off the metal fins inside the unit. You can also use a soft toothbrush to clean the fins of any dirt being careful not to bend any of the fins. You could possibly have a dead animal in your duct work somewhere.

You have dirty sock syndrome.

A very light mix of bleach and water sprayed on the coils will do the trick. Remember very light mix so you do not stink out the house. You migh have to do it 2 or 3 times to kill all the bacteria.