Oddly enough, I think the reason the Jets declined to offer Austin has been overlooked. I think the Jets wanted Austin, and decided they wanted to extend him an offer, but they discussed it with the Cowboys and found out they would match the offer, so they didn't make the Cowboys bid against themselves.
If you think back, when Sean Payton left the Cowboys, he tried to steal players away from the Cowboys, and Parcells took him aside and said (paraphrasing) look, son, its Ok to steal Cowboys players, but there is a right way to do it. You made an offer to our player, then we matched the offer, so essentially you cost US money by making the offer, and didn't accomplish anything on your end. The only person who won here was the player.
The protocol in this case might be different since the Jets coach wasn't stealing a player from his prior team, but this might explain the Jets' failure to OFFER Austin.