I know better now.
I ordered some seeds, from a company I've ordered from before, and planted some in small seed starting containers. I did notice the seed packet didn't have the Latin binomial for Tarragon, but I figured oh well, I'll check the variety later on.
I now have four little four-inch pots of these perky Tarragon starts, so I went searching for the correct name, thinking I'd look at some information before planting these in the ground.
The website said the variety was
Artemisia dracunculus "Sativa". So, happy gardener here googled that up to have a look see.
Found out that Sativa doesn't produce viable seed.
So, I looked through a page of Google images and found that the one that is propagated from seed is actually
Tagetes lucida, or Texas Tarragon. Which is really the plant I'd remembered having once before, and wanted, for the yellow flowers and wonderful scent (though it's not considered "appropriate" as an ornamental by some.)
I sure hope that's what I actually have. I called the company I got it from, got a message asking me to email or LEAVE a message (no hold for the next free representative there!) So, I hung up and composed and sent an email, with the link to the Missouri Botanical Garden's website, where I found the info that French Tarragon does not produce viable seeds, and asked, what do I have?
I then called back and this time, got a live person, but she knew only what it said on the website. However, she assured me she would respond to my email as soon as she had an answer.
I won't buy seeds from them again, that's for sure.