I'm eager, but not foolish. I have just bought my first home, and have minimal experience with home improvements. Almost all of the projects I've done in the house has been with the help I've gotten on forums like this.
I have never laid hands on anything to do with soldering, a blow torch, etc. I'm all for giving something a try, learning as I go. But playing with fire just feels out of my comfort zone.
Thank you for the suggestion of the packing nut. Perhaps I can shut off water to the house, remove the sink and access the top of the valve to tighten it as you suggested.
The valves are both still leaking now with the water turned on - it is a slow leak, just odd drips every few minutes. I suppose I'm going to have to see just how eager (and brave) I am.