From the way you describe it, it sounds like it's a regular gate valve. You might be able to repair it as you describe by removing the packing nut and replacing whatever is broken or stripped out inside.
The only exception to that would be a ball type valve, but those will just have a handle that makes only a 1/4 turn. Those generally aren't repairable, but then again they almost never fail either.
Personally, I would just remove and replace the gate valve with a good ball valve. Gate valves have a tendency to rot out or seize up from lack of use. I actually had one once that disintegrated in my hand when I tried to turn it off (spraying water everywhere and making a nice mess). Ball valves are just much more reliable.
Water pressure:
This could be caused by higher than normal water pressure, that's why the noise changes when you turn on the sink (you reduce the flow to the shower).
Solution:
You could adjust the Pressure Reducing Valve for the whole house (if you have one), or install one on the line to the shower.
Reducing Washer:
In most newer shower heads, there is a washer that reduces the water flow to save water.
Solution:
Removing this washer and replacing it with a normal one should fix the problem (but then you'll be "wasting" water).
Diverter:
This might also be caused by the diverter (the mechanism that diverts the water to the shower head, rather than the tub spout), if the valve is not functioning properly it could cause noise as the water passes through it. This may be accompanied by a small amount of water still flowing from the tub spout when the shower is on (basically it could be creating a "water whistle", similar to how a flute works but with water instead of air).
Solution:
Replace the diverter.
Loose Pipes:
The noise could be increased by improperly secured plumbing, as the pipes are allowed to resonate freely.
Solution:
Properly fastening the pipes using clamps, may reduce the sound.
Best Answer
It sounds like you have more than 1 bad valve. The valve to the home is leaking and the ones to the shower are leaking. If really old you might turn a bit harder for a gate valve, if that won’t shut it off open a hose bib they are usually the lowest point on the system. Then pull the faucet stems and replace the washers. After replacing the washers on the shower I would be checking the shut off to the water heater also many have a cold water main shutoff at that location change that washer also. If you can shut off at the street then change the washer at the house shutoff and you should have fixed everything.
I purchased a kit with 2 or 3 different sized washers they look like little volcanos These are healed in place with brass screws my kit came with some of those also.
Over the years I have repaired hundreds of valves mostly with the washers and some valve stem packing I replace them and still have the metal box (tells you how old it is)
Every home owner should get a kit like this only 10-20$ and start adding as you do other valves, mine also has orings for washerless faucets now you can fix the valves in just a couple minutes in most cases for under a dollar.
Worst case you may need a seat cutter / reamer but most of the time a new washer will do the job.
I am sure there are videos online to provide live examples on how to do it.