Rich,
They're neither native or invasive....They're introduced....But since they're non native, DFW doesn't really protect them much. They went from no fall hunting, to 1 bird for to 2 weeks, to 2 turkeys, & 1 month. This is while are turkeys numbers are dropping badly.....Our population in San Diego is 25% of what it was just 5 years ago.....If we ever lose our population (like San Bernardino NF), we will never have local turkeys to hunt again.
We at NWTF can't make anyone not shoot a legal bird....All we can do is suggest that you don't shoot fall turkeys and especially hens.....The drought, predators, and lack of mast has killed plenty.