The vacuum advance may be part of your running issues. There is a diaphragm in the vacuum advance unit. Not uncommon for the rubber to be blown out. These distributors use a ported advance port in the carb for vacuum signal to the advance diaphragm. The vacuum port is above the throttle blade, on the atmosphere side of the throttle blade. So there is little to no vacuum at idle with the throttle blade closed or mostly closed. As you open the throttle, the port starts to see vacuum as the throttle blade opens, the position of the blade dictates how much vacuum the port sees. The vacuum advance unit will advance the timing depending on the vacuum it sees, until the throttle blade is open so much that there is no vacuum signal and the distributor rotates back. That is the movement you see. If you are not getting any movement, either the rubber diaphragm is shot, the distributor advance is frozen or jammed, or the motor is not making sufficient vacuum when the throttle is opening to actuate the advance. So I would start with the diaphragm. Make yourself up a little steel line and fitting to install into the vacuum advance unit with a rubber hose on it. Disconnect the line at the advance unit, install your little tool and gently blow into the rubber hose, If you are blowing through the diaphragm, no resistance in other words, the diaphragm is shot and this is the first thing to get replaced. If you do not blow through it, suck on the hose to create vacuum and see if the advance moves. if it does, the diaphragm is good. If it does not, twist the distributor to see if it is frozen or jammed.
If the advance is good, then you have to see if the port in the carb is clogged. Hook up a vacuum gauge to some port that is full manifold vacuum (vacuum source under the throttle blade). On the later 55-62 intakes, there is a place for the vacuum hose for the vacuum wipers, this is a full manifold vacuum port. Not sure about your intake. See what full manifold vacuum you have at idle, and a little off idle, say 2000 rpm.
Then, connect the vacuum gauge to the end of the steel line at the advance. At idle, you should have little to no vacuum. Then crack the throttle and gradually open the throttle to 2000 rpm and see what vacuum you have. If no vacuum, the line may be clogged or the port in the carb may be clogged or some other issue that will need further diagnosing. that is how I would approach the problem.
