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!