Sounds like the drive gear on the starter is hanging in the up position. You may try lubing the spiral grooved shaft (helix shaft) the gear rides on.
The solenoid is just a big relay. It uses a small amount of current from the ignition switch to activate it. When activated the solenoid transfers the high current across the heavy cables hooked to it.. One of the heavy cables goes to the battery and the other to the starter motor.
No "special" precaution changing solenoid or starter. Unhook battery first and don't over tighten any bolts as they are threaded into aluminum.
ETA -- If the "drill sound" you mention is more of a screeching sound, and the drive gear on the starter is retracting normally, the upper bushing/bearing in the starter is going bad.
A way to check for this is to spray a bit of WD-40 or other lube at the helix shaft where it enters the top of the starter motor. If the screeching goes away then the bushing is bad.