So sorry you’re struggling. One small thing, do you feel you lack confidence at doing parts of your job? I’m referring to the things you said you weren’t qualified to do. I think some people see someone with a degree like yours and assume they know computers inside out and can fix anything. Not saying you don’t have a lot of knowledge but as you said, it’s a broad field.
I’m not sure what to suggest regarding the team (though if I were the boss, I’d be prescribing some intensive team building exercises ), but it’s understandable you’d get stressed and angry at having to be a go between for them. They don’t sound very mature if they can’t just talk to each other. They’re at work, after all, not school.
I think lizardlady made a good suggestion; you and your partner would, ideally, both be responsible for your own emotional well-being. That is to say, it sounds like he needs to find a way of dealing with his anger at your feelings in a more constructive way. As I said, neither of you can be responsible for the other’s feelings but his reactions to your stress are not helpful.
I will apologise in advance if you’re already trying, it’s not applicable to you or you don’t have time outside work etc, but is there anything you do, or could do that would help you alleviate/manage your work stress? I’m not saying you need to destress to placate your partner, this is about you taking care of your health. I would go to the doctor as you were thinking of doing and see if they could do something, at least.