@PotomacBobAM radio waves are more effected at longer ranges because they "bounce" off of the upper atmosphere, that allows AM waves to broadcast over hills, and even mountains. However, depending upon the amount of power (and some other things) you should still be able to hear local AM stations.
FM is basically "line of sight" which is why you often lose FM stations driving through "hill country" or mountains. Some commercial stations (and most EMS) have repeater towers nowaday.
Depending upon the intensity and duration of a "Carrington Event" it could be difficult to receive radio transmissions; possibly equipment could be damaged.
Faraday Cages (or a SCIF), or lead, or lots of dirt should protect your equipment. My grandfather used galvanized steel garbage cans to store both food, a couple of radios, batteries, etc. Protection from EMP, rodents, etc. He used to keep a couple hundred pounds of dried beans, other vegetables, rice, jerky, bullion, spices, AM/FM radios, a HAM set, and some walkie-talkie radios to.
This was mostly in case of a natural disaster, rather than WWIII.
I inherited his stocks, and like him rotated it. (Using the oldest 50 pound bag of beans/rice, and buying new ones to replace them. Although, I "downsized" to 25 or 10 pound bags. I used them in hard times. During the last few years my stocks have been diminished.
Back to your point. Are you trying to use a radio During a Carrington Event or after? Most often these are local events of short duration. A major event, such as a HEMP (High altitude Electro Magnetic Pulse) attack, or an extreme solar storm effecting the entire Earth?
Old vacuum tube electronics, and some modern "hardened" equipment are more resistant to a Carrington Event.
You would have to be more specific. Events are on a spectrum of intensity, duration, and area of effect.
A small town in Northern Alaska, Canada, or Russia, is more likely to be effected, and become isolated. Hawaii was significantly effected by an event in the 1950's.