How can I fix a frost-free spigot that's leaking out of the handle/rod?

Question

I have frost-free spigot that developed a leak this summer that comes from the handle only when the water is on. After trying to fix the problem then turning on the spigot without a handle, it appears the water is coming through the rod itself. How can I fix it?

I think the spigot is some model of B&K Mueller Quartermaster.

I assumed the problem was with a thick rubber washer at the location labeled B. I wasn't able to get a replacement at the hardware store, but the guy suggested to use some "valve packing cord" next to it instead to increase the pressure on the washer so it would seal better, but that didn't work. It appears the water is coming through the rod, not around the area where that washer seals (see the second picture where I totally removed that part).

There's also a weird plastic washer (labeled A) with a soft plastic sealing material on one end. That sealing material has a piece missing, but I'm not sure what that part is doing or if that matters. Or even what it's called to find a replacement.

Here are some pictures:

The end of the rot that connects to the valve looks solid.

Answer 1

After a little more research, I might have found the answer hidden away on the manufacturer's website:

https://bkproducts.com/wp-content/uploads/BK-Products-Sillcock-Identification-9c.pdf:

HANDLE LEAK INSTRUCTIONS: Remove the setscrew which attaches the cartridge to the stem. The picture below shows the setscrew. Turn the stem upside down. There will be a raised edge across the stem base. On the raised edge there will be a small pinhole/weep hole. Fill this with epoxy or a silicone-based material, allow to cure, and reassemble. The second photo below is of the PINHOLE on the stem base. Be sure to reattach the white “spacer” disc with the rubber side facing the stem base in order to cover the weep hole.

Their "second photo" shows the same thing as the last photo in my question. I've confirmed that water will escape from that small hole if I put it in the handle, so that's probably the problem. Previously I thought that hole dead-ended, because I probed it with a wire and it seemed very shallow.

I wonder what changed make this an issue now?

One interesting observation is their first photo also shows a spring that pushes the "spacer disc" I labeled "A" against the rod end that wasn't there when I disassembled it. Is that what's supposed to plug this hole if the spigot was functioning properly?