Replaced solenoid/ignition switch - Starter keeps running

Ok, try this.
Lets test the neutral safety switch first, by trying to start the car in gear. If it does not attempt to start, then you know the neutral safety switch is working as it is supposed to.
The next time the starter stays running, place the car in reverse, or drive. See if the starter kicks off. If the starter stops when you are in gear, then your problem is in the ignition switch. It takes power thru the ignition switch, and thru the neutral safety switch to the starter. Placing the car in gear should open the neutral safety switch and loose power to the starter relay.
If by placing the car in gear, the starter continues to run, then your starter relay is staying engaged by either being stuck, or power from a source other than the ignition switch.
I do see a connection to the starter relay from the Solid State Ignition Module. Not sure if it may be bad sending stray power to the start relay keeping it engaged. But test first before you replace parts.
If you find the starter stays engaged while in gear, then check for power on the terminals of the starter relay. It should have 4 wires, 2 small and 2 large. We know the large ones have power flowing since the starter is running, but check to see if the small ones have power. No power, but starter is running, means relay is stuck. Power means it has to be getting power from that ignition module. Or a crossed wire somewhere.
It may be best to remove the power cable going to the starter and do all of your testing with a meter. Dont need to have the started engaged the whole time. Just keep trying to start it until you have constant power to the power cable at the starter. Make sure it is secured to not touch anything, or sparks will be flying.
Good luck