Radio works but doesn't get very loud

Can try, open fob and use a hair dryer to dry it out, could put it in a box (like a small oven) and just use a gentle heat, I have dried out ECU's like this, hand dryers are great for this. Hope it works

Radio needs replacement.

I think the clue is when it rains?
Is there a multi pin connection plug on the harness that is in a place where its getting wet?
I'm pretty sure if you check/remove exposed plugs/connectors and give the under hood fuse/relay box a careful look you will find moisture in one or more of them.
If there is corrosion on the pins/receivers this will have to be cleaned up.
Basically, there are permanently live pins in the harness at all times and the damp is creating a circuit(or not) to other circuits whether the key is on or not.
Get a tin of WD40 and a hair dryer(to dry the plugs/relays) A shop airline will do.
Make sure the connectors are fully re-seated.
Try and work out what works, it may lead you to only one connector. Then work out how to keep it dry!!
Last thing, is the harness damaged anywhere?

wet vac & hair dryer

Does the engine have a distributor on it ??? if so see if thats wet, you will need to clean and dry that out.

if its fuel injected take a hair dryer and aim it at the injector rail where the injectors stick into the engine and dry that a bit..

if you just hear a clicking when trying to start the car charge the battery to full, all of the items you listed will work with a low battery , but it doesn't have high enough amps to turn over the car .

First, I checked to make sure all spark plug wires were o.k.. There wasn't any visible cracks in the top of the distributor cap. So I took an old hair dryer and plugged it in. Set the hair dryer on the top of the distributor cap for about 10-12 minutes. Thats enough time to dry the inside of the distributor out. Engine starts fine so obviously there is a hair line crack somewhere on the distributor cap.

Your hair drying may be drawing to many amps depending on the wattage of your dryer

it's worth a try.  also remove the plugs and retighten everything and use a hair dryer to dry inside the plugs.  You may only have a corroded ground wire.

When engine wiring gets wet it won't start. I had an old Mustang that I had to remove the distributor cap and dry it with a hair dryer every time it rained real hard. I am thinking it will start after enough time has gone by-I had a car get flooded twice, and first time it took 3 weeks before it would start.