Im just going to throw a few things out that I have experienced in the past with old vehicles not necessarily model a's.
Be sure your cap can touch the rotor. Be sure the carbon piece is in the distributor top and the rotor top is springy.
Check the gap between your rotor tip and the distributor body.
Check the condition of the electrodes inside the cap and where the plug straps/wires attach. Corrosion will cause high resistance that can cause some funky things.
With the top off the distributor and the rotor still in place on the distributor shaft, hold the high tension coil wire just above the rotor and spin/crank the engine over. Spark should not want to jump to the rotor, if it does the rotor is grounded/Bad.
Pull the cap off the distributor and the rotor off the shaft grab the distributor shaft, give it a forward and backward wiggle there shouldn't be any, or very minimal. My A had too much play and would give an erratic idle and hard start sometime. If this is the case it seems it should still try to pop at some point though.
A Different A I worked on some time ago had the timing rod pop off at the steering column. The issue didnt arise until trying to restart. It looked good initially at the distributor until I looked from the other side. Even if your rod is attached still be sure it is in the correct setting for starting.
My A had a condenser fail once, however that resulted in No spark.
There is a possibility maybe that your high tension coil wire is faulty. In situation it has a break in it but when you hold it for testing it works. If you have a multimeter you could check the resistance of that wire. Solid core wire should have minimal resistance.
I had a weird thing once on a car where the coil had a break in it at the nose where the high tension wire plugged in and the spark would jump to one of the primary posts of the coil instead of go down the wire. You can see that in the dark pretty clear usually. you may even hear the pop from the spark.
Be sure your plugs are dry and clean, may as well check the gap while your there too.
Good luck with everything! I can understand your struggles with the cold, fingers just don't like to work under a certain temperature!