"I don't want to just start throwing new parts at it. "
Ahhhhh, troubleshooting first before buying parts? That's the correct approach. Too bad we don't see more of it around here! :)
If you are 100% sure everything else checks out, then jump the solenoid.
I'm not saying that's the problem; the solenoid is making a noise because of low voltage. That is likely caused by:
1. A weak battery (charge it & check it per tip # 49))
2. Dirty grounds/connections (remove the cables & clean the connections)
3. Bad cables or incorrect sized cables (replace them - see tip # 41)
4. Bad ground between the starter & the block. (pull the starter away from the block & clean the mating area - see tip # 36)
Jumping directly to the starter is also a good test. (but see tip # 43 first)