1984 was either the last year for RWD on the Fleetwoods or the first year of the second downsizing and conversion to FWD. If you have a RWD, its probably has a carb, if you have a FWD, it is definately fuel injected. To further confuse matters when they did this, Cadillac made a FWD Fleetwood, and renamed the Fleetwood Brougham to Brougham, what a bunch of dumbasses. They corrected this in 1993 , it only took them 9-10 years. Then the same fools discontinued the Fleetwood after 1996 so they could make more chevy trucks
Now they are converting everything back to RWD
If you have a carb, it might need 2, 3, 4, or 5 pumps of the gas pedal after stting more than 24 hours, but that should be all. I strongly suspect a sticking choke. I had a bad choke on my 1970, and cold starting was a b!tch.
Even after rebuilding the carb, the choke shaft was still binding. Here's how to check. After a night sitting, open up the hood, remove the air cleaner top and look at the carb, you should have two choke plates and they should both be open i.e., sticking up and down. Go inside the car and press the gas pedal to the floor twice not too fast and release it. Then go look at the choke plates. They should now have shut. If they didn't yoen your choke is sticking from who knows where, could be the spring, the plate shaft, the linkage. Try wiggling the linkage a little, it's usually on the passenger side. You can also try pulling off the throttle (that's the lower plates, which you can't see. The gas pedal controls these and it's on the drivers side and should be a long rod that connects with a linkage to the carb. Move it back to free up some cams (that sit on the other side and don't want to move unless the throttle is moves part way), and then try to move the choke plates. If you're lucky, they will close for you.
If this works, then the car should start. If you can get through this,,then you probably need a serious carb cleaning , 2 cans of gumout spray all over the carb, plus 2 bottles in the tank. That MIGHT clean things up enough, then again you might need the carb rebuilt. There should be a carb shop in your city that can do this for around $100.00.
Good luck and let us know how hings turn out.