help! drake elm is reproducing like crazy

I think the title says it all. I have what I'm pretty sure is a drake elm in my back yard here in a north FL subdivision. It is a younger tree, the trunk is probably 8" diameter.
The last year, and especially this year, my entire back yard is blanketed with little baby drake elms. It is literally like a carpet. There are probably thousands of them. Nothing like the occasional oak that you'll see springing up from an acorn. I've been researching and it sounds like the drake reproduces by seeds.
Anyway, how do I kill these things without killing my St. Augustine grass? It is literally impossible to pull them out by hand. It would be more effective to just till up the yard and re-sod. It is literally that bad. I've never seen anything like it. Can these things survive without leaves? I thought about letting them grow to several inches and then decapitating them with the mower. Help!
Thanks!
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snipped-for-privacy@yahoo.com wrote:

That's what I do with the silver maple that blankets my yard each year. The determined ones get a shot of broadleaf weed killer if they persist.
Dawn
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