Tom-
If your car does not have a thermostat in it, install one. This will aid in the warming up. If it does have one but still takes forever you could try blocking off the radiator partially with a piece of cardboard. If the car starts running too hot just reduce the size of the piece until you get it right. Another thing I thought of is that you might have quicker starting if you change the oil in the dashpots on the carbs for thinner winter oil. My brother and I used to use automatic transmission fluid in them in the winter. Of course we were young and ignorant but I remember it working well.
Your idea that the choke is not being fully engaged has merit. To check it, pull it out and then go under the hood to the linkage and try to move it further manually. Your answer will be self evident.
You could also try adding some "Drygas Additive" to the gas.
LCJUTILA