Q-jet chokes can be a PIA, but when they are adjusted right they work very well. If the electric choke coil is bad, that'll have to be replaced. Once that's replaced, you'll have to adjust the choke in it's housing (all Q-jets can be made to be adjustable, emission models had rivets that are drilled and replaced with screws).
As others have said CHECK your primary pull-off. This is the MOST commom problem with these chokes. Make sure it holds vaccum (a pull off is much cheaper than a choke). The pull off has to be adjusted as well. When the choke is engaged before the car starts, it will close all the way shut. Once it's started the pull off will allow the choke to open a certain amount. If it opens too much will cause the car to run lean, and it will stall out when you put a load on it. Too little, and the car will chug, run rich, and possibly have black smoke.
This can be set by trial and error. Typically, I will eyeball it to about a 1/16 gap or so. Then I will start the car STONE cold and fine tune it from there. Sometimes it take a couple of mornings to get it right.
Also check your fast idle adjustment, make sure you set the fast idle speed. There is a screw on the linkage near the fast idle cam.
Adjusting chokes can be an art, but it can be done by trail and error. It's amazing how a few simple adjustments can make such a big difference. A co-worker had a 78 Corvette that he had to "warm up" for 2 minutes in the summer or it wouldn't drive. I told him that was unacceptable and I'd repair it for him. Two days later the car would start, and could drive away without ANY issues. No parts were faulty, nothing was replaced, it was all choke adjustments.