I phoned a friend whose cb350 put him thru hell with these issues:
"What’s the test light connected to? One end should be to ground, the other to connection to coil primary. With points closed, current flows through the coil, energizing the secondary, when the points open, the magnetic field in the coil collapses and dumps its current through the secondary to the plug. The light should only come on when points are open, as the battery voltage is now energizing the bulb, not the coil.
If the light is connected correctly, it’s staying on might indicate a short in the coil primary, or a poor connection to the points or the points shorting out against the plate. Use a dmm to check that there is correct voltage when and where you need it.
The cb often had issues with the connector shorting out to the cover or the leaf spring on the points getting pushed back and shorting out on the plate."