Sorry that I have taken a while to update everyone on this.......
I have good news and bad news. The noise is "almost" gone. The additional welds did stop the nose for the most part, but there was still an occasional high pitched ringing that was occuring. When I took the auger to the fabrication shop, I did not remove the sprocket and large bushing, so there was still a section of the flight that was overhanging the shaft past the last weld in the back. I could flick this section like a guitar string, and it would also ring, so, even before reinstalling the auger, I suspected that I might have a problem. The sprocket and bushing come off pretty easily, so if anyone else were to want to do this, I would highly recommend it. It will make welding the flight to the shaft much easier.
Anyway, I made another video to explain all of this. Sorry that I'm so long-winded. Again, I'll let you know how I make out. BTW - I used JB Weld due to the fact that this was at the very back end of the auger. I would never use it or anything other than a solid weld if it were any farther forward, as the heat from the fire box would be too much.
Nice work and thanks for posting all of this. Not to be a downer, but would it have made sense to buy a new auger and determine if this same problem exists, then do corrective action? The ONLY reason I say this is the tip of your auger looks about worn out. Could be the picture, but 5-6 year old auger with a stove that possibly had the lower holes on the burn pot might be a candidate for a new auger.