No they couldn't. The 12 mile limit is only for navigational purposes. The US claims a 200 mile exclusive economic zone, and all minerals out to the continental shelf. The states just own the minerals out to 3 miles, except for Texas which claimed 10 miles back when it was the Republic of Texas.
The Truman administration stole Texas' offshore mineral rights between 3 and 10 miles offshore back in the late 1940's, and the federal courts allowed that action to stand. Governor Shivers endorsed Republican presidential candidate General Dwight D. Eisenhower over this issue. That was the beginning of the shift towards the Republican Party in Texas. Eventually Texas' rights to a 10 mile limit were reestablished by federal legislation in the 1950's.
I’m being quite picky, and I’m sure you know this, but the Texas state waters extend out three leagues, which is actually nine miles.
Your point about the economic zone is accurate, although when it conflicts with the economic zone of another country, the boundary is the middle point, which is why Cuba can drill near the Florida Keys.