Options for new equipment when replacing older parts that are end of life? I have a 2004 Giant OCR Touring. It is a used bicycle.

These old touring bikes are a mix of mtn-bike and road bike parts. Most typically a road front derailleur/crank and a mtn-bike rear derailleur. 

The previous owner put a triple compact mtn bike crank on it and it has had shifting issues. Partly due to the FD is braze-on and fits way above the largest chainring. 

The original Shimano 105 FD finally took a dive and broke. It was a 3x9 derailleur.

It still has the Shimano Deore LX rear derailleur and that is running fine.

I am looking to replace the derailleur, and possibly update the crank as well. This is where my problem lies.

The only new 3x9 FD and crankset from Shimano is the Sora. The original crankset was a 52/42/30. These from what I can tell are no longer made. The new Sora 3x9 crankset is close at 50/39/30.

Can I replace my old Shimano 105 FD and ghetto crank with a new Shimano Sora FD and Crank and keep my current RD?
The short answer is yes. 

As @Criggie notes in his comment, the rear half of your shifting system is working. So there is no need to mess with it. 

The front derailleur and crank are both Shimano road product, so if you replace both from current 9 speed triple Shimano road product, you're pretty safe. 

The 9 speed part only matters to maintain chain rings that match your existing chain width, and 10 speed rings will work reasonably well also. 

Do not use a 10speed front derailleur.