I feel like there might be something wrong

Elis, I have similar weather conditions in the spring and fall. Had a 318 with dual exhausts and it acted the same as yours does. It took a long time to get the exhaust system to dry up. The colder the air temp the longer it took. Some times I had to drive it for 15 minutes using lots of throttle to get the water out of the exhaust system. I did this to get the water out before I parked it. [for a week or more]. The black soot was common in my engine, too. When the water builds up on the next restart, it turns the soot into a liquid, and spits it out when you jab the throttle. All normal.

'Smokes black when cold' is likely a choke that is out of adjustment. Pull the air cleaner and plug the little vacuum hose to the air cleaner. With a cold engine, start it. It should rev kind of high. Kick the throttle down to lower the engine speed. It should idle smooth. If any black smoke is seen out back, open the choke plate a little and the smoke should stop. There is an adjustable rod system there. Either a screw or bendable rod could be used. Make a small adjustment and test it after the engine cools down. Repeat.