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You may also have installed the wires on the starter solenoid incorrectly. Maybe not the case, but recheck your wiring schematic and your install work.
OK... Sounds like the solenoid is getting fed through that resistor. Not a good deal. There should be a direct line from the ignition switch "start" contact to the solenoid 'S' terminal and a line from ignition "run" contact to the resistor. Then, the solenoid 'R' terminal and the other side of the resistor get run to the positive (+) terminal on the coil. That way, the coil gets 12 volts when the engine is starting, then 8-9 volts when the engine is running. I would bet that you find the 'outlet' side of that resistor is connected to the 'S' terminal on the solenoid, and that [high] current draw is overloading the resistor.
Yes. If the magnetic 'pull' of the coil is not great enough to overpower the return spring, that is exactly what will happen. And, even though it won't activate the solenoid, significant current to 'feed' it will be flowing through that resistor. You really need to chase down [and mark] the existing wiring and compare it to the schematic for your car. There are some wires run incorrectly, I believe. Here's how it should work:
"Start" 12 v. signal from ignition switch should be run to the "S" terminal on the solenoid. "Run" 12v. signal from the ignition switch should be run to one side of the resistor. "R" terminal on the solenoid switch should be run to the (+) terminal on the coil. Output side of the resistor should also be run to the (+) terminal on the coil. The (-) terminal of the coil should be connected to the wire coming from the distributor.
If you correct your wiring to provide as listed above, the car should work just fine. {crossed fingers} At least, that's the best I can do from a keyboard to help you.