I think I might have an engine problem... :(

Sounds like a big cold rich engine to me, trying to flood itself to death,lol.
Yup. Not trying to hijack but my 408 did a very similar thing last Saturday. Ran very rough at start up (manual choke) and kept loading up and dying. It always runs rich at idle and has to be babied until warmed up (it was 47 degrees F). I have a couple of plugs that come and go and a buddy noticed that one was back feeding up the spark plug wire and jumping over to the next wire. Pulled a couple plugs and they looked very sooty for having 500 miles on them. A fresh set of plugs and $25 worth of non ethanol premium fuel made all the difference in the world.