I am not sure this is a starter solenoid problem. If the starter solenoid was stuck in the "engaged" position, then why would it quit when the clutch pedal is let out? Suspect the starter disengages because the clutch safety switch interrupts the current to the starter solenoid. The starter continues to run for a very sort period because of the pull-in/hold feature of the solenoid.
Regardless, it is easy to confirm what the problem is with some tests. Test for +12 volts at the starter solenoid. Test for +12 volts at the clutch safety switch. Post the results.
To the OP, how much of this do you want to fix yourself?
If it turns out to be the ignition switch and you wish to drive the car to the shop, here is how to do it. Make sure the parking brake is set and the transmission is in neutral. Disconnect the BK/W wire at the clutch safety switch.
Put the ignition into the "run" position. Use a jumper wire from the battery plus(+) to the BK/W terminal. Once touched, the starter will engage. Once the motor fires, remove the jumper.
With the BK/W wire removed from the clutch safety switch, you will not have to worry about the starter attempting to engage while driving to the shop.
Or the solenoid wire can be removed and the car bump started.
If you wish to fix this yourself, start by disconneting the solenoid wire at the starter. This will allow work to proceed without risk.
The ignition switch can be removed fairly easily. However, it seams silly to post the procedure unless it is needed.
Also, consider getting a Chilton's service manual. It is well worth $20.