I need some ideas on where to start looking with a problem I'm having with my 1911...

When the hammer is cocked and the thumb safety is engaged, if I depress the grip safety and pull the trigger, the hammer will drop (it'll either drop all the way or at least drop half way).
Obviously, this makes the gun unsafe to keep as a HD weapon or truck gun, but otherwise operation is fine when just on the range in controlled circumstances.

That said, I haven't shot the thing since discovering this problem and wanted to get some ideas on where to look and what to look for.  Looking at various diagrams, I'm thinking that either the sear is worn, or perhaps the safety is worn and not fully engaging.  I suspect this because this ONLY happens in combination with depressing the grip safety and pulling the trigger,  If I just pull the trigger without pressing the grip safety, nothing unusual happens.  Likewise, if I depress the grip safety and violently shake the gun, it never drops the hammer either... so it seems to be a very specific circumstance causing this.

So, any ideas on what I should look at first, or if I'm on the right track suspecting a worn sear or safety?

It may be the grip safety. It may be the sear. It also may be a host of other issues. One thing it definitely is though, is an UNSAFE gun.

Either way, get it straight to a quality gunsmith and have it fixed appropriately.

Then ask the gunsmith what the problem(s) was.

Yep, I agree, it is rather unsafe, which is why I haven't fired it since finding this problem.  And yes, I can take it to a gunsmith, and most likely will, however, I'd like to try to sort this out on my own first.  I'd like to learn to repair firearms myself.  Most of what I own are fairly simple and I can strip them down to parts and re-assemble them without a problem.  This 1911 though, has some difficult to reach parts, and while still not a terribly complex mechanism, is still more complex than some other pistols I own.

So in any case, the point is that I'd like to learn how to do this on my own first, for my own education, because I am mechanically inclined and enjoy fixing things because there's a certain satisfaction in being able to say that I can work on things and make them work again.

That said, without stripping it completely down I still haven't figured out if its a worn sear or the safety... but I did see the sear slipping and pushing the safety down into the fire position when pulling the trigger and manipulating the hammer.

If you're mechanically inclined the parts in the deep dark recesses of the 1911 are not that difficult to get at.  I recommend finding an online tutorial for how to fit a new thumb safety and do all the checks required to fit a new safety and see if that sheds some light on your issue.

The thumb safety is not fit correctly.   A 1911 sear does not wear in the sense that it gradually becomes unsafe.   A properly fit thumb safety should allow zero movement of the sear when the trigger is pulled.   You can remove the grip safety, reassemble the gun, and cock the hammer in order to see what's going on.  The lug of the thumb safety, when engaged, should block the lower part of the sear from pivoting back.   If yours does not, a new thumb safety will have to be fitted to the gun.

It sounds like the grip safety, at least, is working correctly.

Does the thumb safety go to the "off" position when the hammer falls. A properly fit thumb safety will not allow the sear to be moved partially or fully off the hammer hooks. The more common failure of this type is that if the safety is ON , the trigger is pulled the hammer will not fall until the safety is taken off.
Let me ask a question, Can the thumb safety be operated to its full position with the safety ON. if not you may just nee to fit the existing safety.
With the pistol unloaded put  the side of your thumb in front of the hammer. Pull the trigger and see if you can detect movement of the sear coming off the hammer hooks. If you do replace the safety. The new safety mostly likely will need to be fit. It must fully engage but not allow any rear movement of the trigger/trigger stirrup.