The resistance coil is embedded at the bottom in some kind of ceramic cement but usually you can remove the small burned segment going toward the brass. It may break off, high heat seems to make the wire brittle. Try not to disturb the winding that is still OK. The repair idea is to put a U loop of copper around and through the last 2-3 turns of the resistance wire. Dress it carefully so the scraping contact on top won't hit it or cause a bump. The repair strand is from 12 stranded building wire. It goes to the brass. In this case it went toward inside then outside.
Now there is a pocket left where the burned part was removed. Since solder will not “take“ to the resistance wire and we want to fill that pocket, we make a filler mesh of the same strand folded tight about 4 times and 1/4” - 3/8” long. Stuff this into the resistance wire coil so the end is in the space. This will serve as a scaffold for the solder to tin and helps contact too. Solder shrinks a bit as it hardens which will grab the resistance wire and give us a good bite. Scrape the resistance wire with a box cutter to get it shiny at the end coils.

Now set it up vertically, use a good soldering iron (I use a 250 watt gun) and solder all this together. Try to leave the top of the solder level with the brass. You want to make a smooth ramp and not leave a big lump of solder although you can file to shape later. My switch had a notch in the ceramic and I ran U ends out soldered to edge of brass. Do not get solder under brass as it will tilt it.

Check the resistance wire with an ohmmeter. To the end is 10 ohms and at the brass end it is zero ohms.
Reassemble.
The spring goes in the ceramic pocket (I put it on wrong side of pronged plate first time).
I noticed that two of the 3 copper rivets support the sliding center contact and I connected them together outside (black sleeved wire). Wow no flickering!

The short in my switch was at the pod the switch is in. The wire from the rheostat to dash lights was scraped and touched the inside of switch pod. To prevent this from happening again I added a 3A fuse. On a 300F this fuse might be a 5 amp due to EL lighting . It is easy on this C with the external wire feeding the rheostat. On an F you might have to put the fuse in the orange leads splice together to one wire onto dimmer terminal.

While doing this project, I discovered the drawing of the headlight switch in the 57 factory shop manual has an error. Talk about maximum confusion! The 57 manual is below on the left. See how the feed to the headlight switch comes to the top where the 20 amp breaker is located. This is wrong. The 58 manual is on the right. The feed comes to the B terminal. This is correct.

The big takeaway is the 20 A circuit breaker with big stud on it is not the power in or out. It is only live when headlights are on and cannot be used as a 12 Volt tap or for a battery feed.

The two sockets marked "IN" are exactly the same connection. They are the instrument feed and tailights. The two sockets near the metal stamped "P" and "IG" are front park lights only and possibly the ignition key light on other models. The 300C does not have a light at the key.

"B" is battery feed. "A" is accessory feed constant 12 Volt for the brake light switch, the clock, the dome light and the map light. I believe there is a 20 Amp breaker inside the switch between the "B" and "A" terminals. Headlights are fed from the "H" socket.

A final thought on all this. As an option, you can rewire the orange dash light feed into the tail light connection which will bypass the dimmer and give you full on dash lighting.