Jay',
First, get a copy of, or look in, your Austin-Healey service manual. Be sure you get the BN-2 supplement because there are some differences between the BN-1 overdrive circut and the BN-2 system. There is a simple and comprehensive troubleshooting guide you can use to pinpoint the problem.
Second, there are two coils (a closing coil and a holding coil) in the overdrive solenoid, both are energized initially and then a set of points is opened when the core
is magnitized, moving the plunger, and cutting out the closing coil so the current requirement drops off. It keeps the coil from overheating. If one of your coils is shorted the solenoid would be weak. I would not put 24 volts through your solenoid again. I work with 24 volt systems in aircraft. The speed with which smoke can be released from wiring is frighteningly faster with 24 volts than with 12. It is designed to work on 12 volts. It must be strong enough to move the lever on the gearbox. I can't tell you how much force that takes, but if it doesn't move the lever you probably have a bad solenoid (assuming no problems inside the gearbox). A good loud clack is what you should hear when the solenoid is engaged. If the points are not closing properly you would only have the holding coil trying to operate the magnet, again, it would be weak.
Hope this helps.
Jon Robbins
Jon Robbins