I don't understand why you wouldn't just go back to a MTL setup, at your former VG/PG mix? I agree with everyone saying that also, maybe sub0hm devices aren't the way to go, as it's far more vapor being inhaled, and typically folks do use more VG in their mixes. You can just buy a bottle of PG and dilute your current juice to see if that WORKS. Yes you'll have less nic, but it would mainly be to find out if that is the issue.... There are tons of vendors who will mix you the VG/PG strength you desire, at whatever nicotine strength you desire. You can also DIY juices at a lower flavoring percentage, if you are worried about inhalation issues. I think you just ran into a setup and ratio that are messing with you.
I couldn't use PG almost at all, I'm sensitive to it on inhale, but I went to 100% VG juices and it was fine. I DIY so I have control over my ingredients, and I have no symptoms of anything....
If you want to try the gum, that is obviously up to you, but had I let my early experiences stop me from figuring out the problem, then vaping, I'm fairly certain the gum wouldn't work for me, traditional nic pharmaceutical options have not tended to work for me. But, you could try it. It's expensive, you'll at least be chewing something but you won't get the same hand to mouth action.
I *do* understand being worried about vaping giving chest sx, but it is not at all uncommon, and there are remedies for it, so it may be worth trying a MTL setup (there are still some great new options out there) and lowering your VG. Of course, I am not a doctor nor a scientist, so you are going to have to do what is right for you.
Anna