What Jamie-Dog-Trainer said.
I would NOT use a Stay command. When I say Stay it means I want my dogs to FREEZE IN THAT EXACT SPOT until TOLD OTHERWISE. But then, to me, Down means Down till I say move, but they can lie out on their sides unless I say Stay.
When I leave, my dogs get a cheery "Be a good dog and guard the house." And sometimes a stuffed Kong or a filled Buster cube. Something special they don't get when I'm home.
I don't like the current feeling that everything a dog does is dominance related, but in most cases of separation anxiety, unless you have an extremely servile, almost fearful dog - the kind of dog whos owner carries the dog's confidence around, if you know what I mean, this usually is a dominance thing.
Compare it to this, when you were a kid of 10 or so, and your parents left, you didn't freak. You may have been upset for a minute that they didn't take you with them, but you didn't freak out the whole time they were gone.
But once you became the parent of a 10 yo, if that kid were to up and leave and you were locked in a room where you couldn't go after them and had no idea where they were, who they were with or when they'd be back, you would be OUT OF YOUR MIND!
OMG, where did they go, anything could be happening and I'm not there, how could they just up and leave like that, what if there's a....
See what I mean?
So while I'm sure that as far as Jax is concerned, the sun rises and sets around your DH, I am equally sure (even though he may obey commands readily) that Jax has tough time thinking that your DH is capable of taking care of himself in the Big Ol' World without capable Jax right there, just to make sure.
So maybe have DH work on the "little things" in the language of Alpha. Body language, eating first, through a door first, not complying with demands for petting (ie; dog comes up and nudges, don't just comply with an ear scratch, do a puppy push-up Sit, Down, Sit, and he earns the ear scratch) that sort of thing and see if that helps.
