You might be getting too much full advance timing as mechanical advance is in addition to the vacuum advance (under certain load conditions) - have you checked with a timing light? It takes 10-15 inch of vacuum to pull the vac advance - you cannot create that much with your mouth. There is also a spring inside the vac advance that controls movement vs vacuum. This can be adjusted with an 1/8" allen wrench thru the port. I've had a similar issue you are noting - mine was caused by damaged wires at the module - every time it moved with vacuum advance, the wire(s) would lose contact and engine would begin to miss until I let off the throttle and it returned to no advance position. These can be finicky to troubleshoot as more than one issue may be present. You will need to isolate the issue first. Try w/o vacuum advance hose first and check full mechanical timing at 3000rpm. Then connect the hose, and repeat. Anything past about 35 degrees total advance is going to cause hesitation or backfire.
The other high probability cause of hesitation (for carburetors) is accelerator pump on carb is not working. Be sure to look at that too.