I don't do tanks, but isn't there a removable piercing plate? Do people remove that when cleaning the atty?
I haven't tried this yet on the primer thing, but it ought to work: If you remove the wick-plate, and take some napkin to that silver part around the coil (a lot of juice collects there, so I imagine primer, too), you could mop up the excess. I usually just vape on through it.
Many people, I've read, seem to not remove the wick-plate when cleaning, which is understandable... but to do the best cleaning-- the dry-burn-- you kinda have to, so you don't fuse the wick to the atty (the darker juices tend to caramelize on the wick, and this fuses to the build-up on the coil, leading quite a significant drop in vapor production in a hurry...)
It does require a little tinkery-do to get the plate out and in, but there's vids and whatnot on that, and IMO it's worth it, for the extra life-span you get from the atty. Once you get the hang of it, and find some tool which will work for it, it's not a big deal. I use a little hook thing-a-ma-jig that I found around the shop-- captain obvious reminds you to make sure whatever you use for it isn't rusty!