I searched other posts here, and found similar issues, but I haven't been able to find exactly my observation.

I have the apparently very typical "compressor clicks" observation, but when I spin the fan with a screw driver after it clicks, the fan just spins free a little bit and doesn't engage a motor or anything. I'm mechanically inclined and with time can manage to figure most any problem out, but I'm here visiting family so my time is limited and its raining out today.
My gut says the fan motor is gone, but as I have zero experience with any type of residential air work, I hoped to find an experienced friend here to lead me.

Thanks a ton for any help! Even with the rain, we're still pushing 80+ degrees here in Mississippi.

When you say the compressor 'clicks', do you mean it is running correctly, or just trying to start but not starting? I would start by checking the capacitor (most likely a dual cap that runs the compressor and fan motor). Caps are cheap and easy to replace, so a good place to start. Many times if the cap is completely shot on the fan side, the fan won't even start with a push start, just dead. Let us know what you find.

Sorry for the vague description. I have a window right by the outside unit, so when I'm out there, I have another person in the house switch the tstat to cool. When that is done (I can't see them, so I don't know if there is any time delay) I hear a "click", and that is all; other than the very faint hum, that my limited ear would compare to a seized (small-because of how faint it is) motor. Like I said though, the fan moves freely.

I just went out and pulled the panel when we got a little break in the rain. The model is a trane xe1000. I noticed a small red led blinking (I've flipped the outside breakers off) and I noticed one of the hot in's (red) were melted and corroded on both sides of the wire nut. I don't have any heavy romex on hand to replace that immediately, but since it seemed to be working last night, and there's corrosion on the exposed wire, hopefully we can diagnose a problem beyond that---even though it would be amazing for it to be something that simple.

OK, an update that may spark a thought.

I turned off the outside breaker when I took the access panel off to take a peek. I just thought about going to see if the contact pulled in when the tstat was switched to cool after a long day at off. It in fact did, and that is apparently the click I was hearing.
The power is still off at this point, so I thought for the sake of experiment I'd throw the breaker to on. When I did, both the fan and compressor tried for about .9 seconds to start, then just back to the faint hum I described earlier. So, at least momentarily, the fan got power to spin it, and the compressor got power to try to start it.
Is there any chance that points to something obvious to you guys?

Thanks!

In the way of closure for any future references, I went back out a little bit ago and poked around some more.
I fixed the burned a/c (not to be confused again with ac) feed I mentioned earlier, and tried the contact test again. This time I still heard a click, but I didn't see the actual contacts move like I did last night. Hoping for the best, I pulled the contact out after calling around for a replacement. When I got it out however, I noticed what looked like 6 little legs around one of the two contact points. One paperclip later, I removed some kind of small insect that was in the wrong place at the wrong time two nights ago. "skeptically optimistic" I put it back in before going to get the replacement, and all is well.

Thanks for the update. Always appreciate that. I was going to guess bad dual cap before your update (which will cause the same symptoms you saw). However, it sounds like you found the culprit. Im betting it wasn't letting both contacts , well, contact. Allowing only half the voltage to the compressor and fan motor. Good catch, you saved yourself some bucks. Well done.