I'm afraid that I'm going out on a limb here by trying to
answer you when I'm not sure, but I see that no one else has
replied to you yet.
I don't know much about visa sponsorships and the like, but I
can tell you a few things about DE and hiring.
Here in DE, we do not hire by county. I know that plenty of
American states do, but Delaware does not. Instead, they hire
by school district.
You'd need to look either at http://www.teachdelaware.com or
at each district's page in the state; many of them have their
own employment website, and they definitely do not all use
teachdelaware.com. For instance, Caesar Rodney School
District has their own employment page:
I wish you the very best of luck.