I threw a set of these Dunlops on the thing sometime in 2019 -

https://www.goodyear.com/en-US/tires/dunlop-sp-sport-maxx-tires

I thought these were supposed to be good in the rain? I didn't go cheap on the tires.

That tire appears to be designed primarily for dry summer high performance driving.
The center rib is pretty much solid with some sculpting in it.  That makes steering response quicker when its hot and dry.  The drawback is that near freezing and rain/snow will allow it to hydroplane at interstate speeds.

I’ve had tires like that on a SUV, traction was terrible.
As TKO mentioned, for wet/snow and cold weather around 40F get tires that have more siping.
At lower temperatures an all season tire will provide better traction.

What tire pressure are you running?

I like the “reassuring statement” on Tire Rack.

Years ago I had a set of Continentals on my 86 GT. They stuck to the dry pavement really good. When the road got damp or wet they were like an ice cube on a hot griddle. I got rid of them as soon as I could.

Got to agree with sal.  Mustangs originally ran about 27-28 psi on bias tires. I run 32 on mine.  Try thirty next time just as a comparison.

Wow... 30 - 32 Psi for a 15" tire seems low to me, but ok... I will give that a try.
I had just been going with what late model cars use for tire pressure, because I thought the pressures quoted in the 60s were for cross-ply tires.

Just to add..............
I have "neglected" checking my tire pressure on several occasions on the 'old-hot-rod' and have found the pressure to be in the loooow twentys!! They looked pretty good butt pressure was low. I guess........because the car is so light, the 225-60-15 hold the car up just fine with a minimum amount of air!?!
36-38 psi is A LOT on these light weight cars (unless you are trying to wear the 'middles' out)
6s6

Yeah, but that 14.7 will only register in a vacuum.

First thing I would do is drop that presure, its way to high, get down to 28 -30 max and try driving it. My coupe seams well footed in the rain as I live in Oregon and that sometimes is our sunshine. 
I have always used the caulk test and then adjusted one way or the other to get the feel just right. The PSI on the side wall is for max load rating.  I run 225/ 60/15 TAs on 7" rims on my  65 coupe and they are at 26. A good example is with my truck tires, C 10 stepside  they say they are 17-5 x 36 x 15s and to get them to be 36" tall it takes the full max 30 PSI, they also max load at 2800 each. truck only weighs 4K   I run 25 in the fronts and 18 in the rear because its a short box.  Anymore and there foot print is 6" wide. I still dont get much more that 10" on the road. You cant drive it in the rain or you will swap ends everywhere.

Mud has no traction, so typically the best mud tires are ones that clean out the tread efficiently to allow you to try and claw your way down to something that has traction.  The guys who run mud bogs use tires that almost look like they have paddles on them.  Its more churning through the mud than biting into anything.  Super Swamper Boggers are one of the most popular for this, and I've seen guys carve on them to further improve paddling.

Snow is a strange thing, because snow traction is best when the tire is packed with snow.  Snow on snow traction is the best you're going to get.  I find mud tires, expectedly, do terribly in snow.  They claw through the snow which just makes you dig holes and get stuck.  A less aggressive tread that fills with snow and stays packed allows you to stay on top of the snow and move along.

One thing is the same about both conditions: once moving don't stop.  Wheel speed can to a point make up for a lack of traction, but its got to be controlled chaos or you end up wrecked. A $500 beater will teach you all kinds of things about traction, wheel speed, etc. if you have a decent place to wheel. 

An update on the tires... I tried running them at 30psi and it was horrible. The car would wander... Handling and driving was horrible, so I am currently back up to 38psi and the car is SO much better. I haven't had an opportunity to drive it in the rain recently, but maybe the Dunlops are not the best in the wet. In the dry they grip really well.
For those of you who run tire pressure at 30psi and 32psi, I don't know how you do it.

I've noticed over the years that a car with alignment that is out of whack will seem to handle better with more air pressure in the tires.  I think Nos681 might be on to something there. 

The alignment was done within the last 6 months by a reputable shop, where the Technician regularly works on classic Mustangs. One side was way out, but they fixed it. The alignment was done for radial tires and the numbers were near perfect when compared to the non-Shelby drop numbers quoted.
I just think 28-32psi is too low. I have 38psi in the tires now, but could drop back to 36psi. I doubt that I would go less than 36psi...