Shed chimney/ "flue" leaking (pics)

Greetings friends: I hope this isn't too off topic for this forum. Here's whats going on: I have a potbelly stove in my shed with a 6" standard flue pipe going up through the ceiling. When it enters the attic of the shed, it is double-walled from there on through the roof. (See pics attached
It has been leaking for a few months now so I went into the attic to take a look and I could see daylight in a few pinhole-sized areas around the OUTSIDE perimeter of the double wall. So, are there any options to patch it with some kind of heat tolerant caulk or do I need to tear up the flashing and redo the whole set-up? The shingles are fairly old and if I can avoid opening a can of worms, thats ideal. Thanks in advance.

The acids and moisture have "used up" the galvanizing coating.

I would paint the top with "stove paint"

Edit:
I would also cut the shingles back about 1/2 inch on the top side so that the shingles are not in contact with the pipe.

Then paint...

vent/flue

If that is conventional class B double wall it is not for use with a wood burning stove in the first place. In fact you should have triple wall pipe with the air spaces between the layers.