I run pipe cleaners through the stem when the pipe is cool, after dumping the loosened ash from the bowl. Then use a folded cleaner to finish clearing the bowl. Dump again. Then lightly blow through the stem, covering the bowl with my palm, to raise the dust. Dump again. Sometimes, I clean the bowl inside further with a paper towel, but rarely. Every week or so, I run an alcohol soaked pipe cleaner through the stem. I've been using Bourbon or Scotch, but may get a dedicated bottle of Bacardi 151 for this. To get Everclear, I'd need to drive to Wisconsin, which is near enough, but Bacardi 151 is probably as good, and is more potent than what I've been using. I don't usually pull the stem and clean the exposed areas. Maybe once a week I do that.
I need to get some Obsidian Oil with my next tobacco order and pay more attention to the outside of the stems. Now I just rub off the grime when I see or feel it. I don't do anything to the outside of the briar except to clean the rim of the bowl. I rub it with my thumb, and, if needed, a bit of saliva. If there's some scorching and it hasn't come off, then I lightly rub a bit of cold coffee on it and this does the trick. I just now dipped the lip of a pipe in some cold coffee sitting here in my cup, and that cleaned the haze and grime really well, so I'll do that unless someone warns of a bad effect of coffee on Vulcanite, as I've read here of alcohol being bad.