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Just checked the wiring diagrams, and it looks like the relay is powered off of the same fuse, no. 11, as I would expect. If they relay is switching the light on the gauge cluster, then you are at least getting power to there, and it sounds like the wiring in the shifter button is fine.
Only two other possible faults remain: your wiring to the solenoid is not continuous (still broken somewhere, or a poor connection is preventing it from energizing the solenoid); or your solenoid is bad. Actually, I suppose a third, but less likely problem could be that the relay is switching the (ground, I think?) for the indicator lamp, but not passing power to the solenoid. You can verify that by back-probing terminal 87 (a white wire) with a voltmeter or test lamp to see if you have battery voltage.
You could also, btw, use an ohmmeter to check for continuity on your solenoid. With one lead on the wire you just replaced and one on a good clean ground, you should be showing only a little over 13 ohms, IIRC, assuming the wire has continuity and the solenoid is good. Infinite resistance (no reading, no continuity) could mean the wiring is faulty or the solenoid is burned out.
To answer your earlier question directly, the frayed/broken wire would do nothing but blow a fuse, not harm the solenoid.
Good luck!
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