You can't kill artemisia. You can cut it to the ground, leaving 1" so you'll know where it is, and it will come back up. I cut artemisia here in central Texas in late August, make bundles of about 3" diameter (the stems) and hang them upside down to finish drying. I've also had good luck just letting the plant dry on its own, and then cutting the stems.
It spreads by root, so expect your artemisia bed to expand.

Hahahaha! Kill artemisia? Not possible. Not even yanking it out "roots and all" kills mugwort!

I find drying herbs on screens rather than hanging in bundles works better in my climate. Try different methods and see what works best given your own humidity and heat.

FataMorgana

I forgot to say that I have Silver Mound Artemisia. Do the same rules still apply? I can cut it practically to the ground and it'll come back?

Thanks so much for your help, by the way.I want to dry my artemisa so I can try to make a wreath. I've heard recently that there are other artemisias that might be better for crafts but I didn't know that when I bought my three Silver Mound plants at a nearby nursery.

Silver Mound is a lovely artemesia. It may be a bit small for making a wreath, but give it a try! If you truly want more artemisia, you might want some Silver Queen, which is taller and easier to bend and attach to a wreath frame.
Drying on a screen is a great idea, and we would do it except the chickens would find it and have a "toss the artemesia" party.
Hint: Never buy three artemesias. Buy one, and by season's end you'll have more than enough!

I started wormwood from seed and now have enough camphory, silver, ferny foliage to stuff a mattress with. The bushes grow to about 4' tall and 5' across, take severe pruning, and self-seed.

Silver mound doesn't get quite as big but if the plants are a few years old, you can cut them back enough to get a wreath's worth. And their foliage is soft and delicate.

Also, you will find wreath making easier with fresh plant material rather than dried. Dried doesn't bend as well and sometimes will shatter. Unless you are making a wreath by pinning bunches of dried material to a foam form.

I was able to make a small wreath with my Silver Mound Artemisia and the plants seems to be coming back even though I cut them pretty severely in order to get enough for a wreath.

If I want to try a different, more productive kind next year (for bigger wreaths, and lots of wreaths), what is generally considered to be the best variety to use? Silver Queen? Silver King? Powis Castle? And where can I find those varieties? I've never seen them at any of my local nurseries.

Mail-order can get you all manner of plant or seed. Search for the named varieties you are looking for at google. I'm sure you'll find a good source.

FataMorgana