I have a '91 240SX S13 DOHC. I finally had to change the starter (original). The old starter had some kind of short, and was pulling in excess of 22 amps. Is it possible that those high amps on the old starter have damaged other parts of the ignition system?

Could you mean 220 amps because a normal starter draws between 180 to 200 amps. No this will only effect the battery cables themselves. The cables take the load when the car is cranked. The solenoid is just a switch a takes very small load. If the battery connections are bad or cable are corrodied internally at the clamp ends this can mislead you into believing you have a bad starter do to high resistance in cables or ends.