No shame to put her down if you have reached the end of what you are willing or able to do.
Esp. if your husband is pushing for it, it is hard to have such a disagreement between 2 spouses. But in no shape or form would I agree to sell her for cheap or give her away for free. I'm sorry to say it bluntly, but that is a cop out. It is difficult for you to maintain her at the additional cost and work, and you HAVE a emotional bond with her.
Someone that doesn't have that emotional bond is not going to try half of what you did. They will put weight on her and send her on her way. Or simply let her decline and most likely die a rough death once they figure out they can't "fix" her.
All that said, I feel like you haven't quite reached the point of letting go. If this treatment plan is all your vet could come up with, it sounds like they need a new job. A dex shot every few month is not going to help her. She could have already been a lot better and managed her weight better with some relief from better treatment. If she has no heave line other than every few month, she is not a severe case. As a matter of fact, it sounds more like a nagging respiratory infection that should have been cultured and treated with antibiotics.
For the vet to say "she's done for, waste of money" because "there is no cure". Well. There is no cure for diabetes. Cushings. And a great many other things where you can still have great quality of life if treated right, whether horse, dog or human. And it doesn't even have to be a expensive treatment either.
Even if it is heaves, why would they tell you to give a shot every few month? You might as well not do it, honestly. There have been a few horses over the years with heaves at my barn. All were managed and rather well. A couple were far more advanced than your mare sounds like. Dex has possible side effects. What medication doesn't though. If it gives her a couple (or more) good years at a very manageable price, what would be wrong about trying? She would probably do rather well at a relatively small but REGULAR dose. Really not impressed with what your vet came up with.
Once she feels and breathes better, it would be far easier to keep her weight up nicely during winter. With very few tweaks to the diet. Thin coat? Many horses need blanketing. Also not a big deal.
I'm saying all this not to make you feel bad, but only because I feel that you aren't ready to give up just yet. If you get to the point that you are, honor her life by letting her die at home, at peace. Not in a strange place, scared or facing worse then euthanasia.
She is young, her disease doesn't sound that advanced esp. given that she isn't on Dex treatment plan. If it was me, I would have a culture taken to see what is really going on. The gurgling and mucus sounds like an infection has settled and she hasn't ever gotten rid of it. A Culture will make sure the right antibiotics are given, as not all of them will work for all respiratory infections.
Sit down with your husband, see if he is willing to get another vets point of view, given on the info you have received from other members on here. You both need to be on the same page. He needs to realize you have to feel at peace with your decision. And if you both are, give her the gift of being let go at home and at peace.
I completely agree with this however, while I would NOT put her on the market I would consider a rescue or experienced person you know as an alternate home.
do you know a lot of horses getting regular year round Dex? I've worked with a lot of vets/horses and dex is either short term as a treatment or as needed for flare ups. I'd be concerned if she needed dex regularly, and I don't think her vet said to do it every few months, just as needed (which for this horse is every few months). It's not just a medication it's a steroid and there's a much higher risk of more serious side effects then your average medication. Oftentimes it does as much harm then good, I've dealt with that plenty it's not pretty and it's usually used for emergency situations or as a last resort medication wise. However, I agree at this point this horse needs more, and maybe one of the things to try ruling out other things (and I agree there's likely other factors) is to up the dex, because to be blunt side effects don't matter if you need to put her down, if that's her last resort then go for it. If dex can buy you a few more years of good qol vs putting her down in a few months then that's all that matters. I do think a different vet that is actually interested in helping you would make a world of difference for her!
Too add- I am NOT at all belittling the care you've put into this horse!! However, I think you've only done the bare minimum (and I think a different vet would explain that to you better). Caring for a horse with heaves can be a lot of work and it's understandable if that's too much for you, however, given a little more time and hopefully finding the magic combination for your mare can make things routine and much more simple/less overwhelming then they are now. Sort of "work smarter, not harder", could be you try a bunch of different things and she is STILL a huge amount of work and not doing well, but since you've barely scratched the surface as far as what you've tried (keeping her out, square bales, these are great but when that doesn't cut it you look at part 2- I don't think you've tried much for medications/supplements? etc). It could be her magic combination of care is right around the corner and you just haven't found it yet. I'm not saying to keep on going indefinitely, but it's at least worthwhile to look into different things. I knew a pony that was kept out, with out being a little dirt paddock next to the dusty ring. I convinced them to STOP keeping him out and got him a corner stall with good airflow but away from the dust and he improved and no longer needed as much care. So as I said maybe give yourself some limits for what you are willing to try and how well she is doing by that time but don't necessarily give up right now. If you can rule out other health concerns and find a good balance (and that may be an if) she could do very well for years to come.