I'm sorry you're upset with your situation - that's a horrible way to feel. You've acknowledged a lot of things you're unhappy with, so that's a starting point already.
1) (from your comment) "I've wanted to email my manager and explain how I feel but I don't really know what I want the outcome to be."
Have a quiet think about this, in your own time. Try and figure it out, think about what you'd like, and what would need to change in your current role for you to be happy. It's a good exercise to do, even if you don't go to your manager in the end.
2) "I have such a good relationship with the users that I almost feel like I'd be walking away from a big family if I were to leave... So much so that even the higher support teams regularly ask my advice on problems that they are unable to diagnose. Furthermore, the company of 600+ 'users' often bypass the usual service desk route in order to speak direct to me."
These are two sides of the same coin, and obviously you want to make your customers/colleagues happy. However, you shouldn't be doing this at the expense of your happiness. You should start training the 600+ users to go through the proper channels. You already enjoy a high level of respect - it is okay to say no. Explain that you'd love to help but you're a bit swamped with other work, so would they mind logging a ticket? It will take a while for people to get used to this, especially if they've been allowed to bypass the service desk, but it will make things easier for you in the long run. Then, if someone gets in touch and you have the time and calm to deal with your problem, you can always do them a favour and bypass the service desk, but it should be a favour, not the norm. There's a service desk there for a reason.
You obviously have lots of knowledge and capabilities, and your users know this. However, that doesn't mean you have time and superhuman capabilities. You need to reassign other items - so what? Guess what, that's part of your job, and the job of the others on your team. Their problem isn't necessarily more important than anyone else's. Get your users to start using the proper channels.
(There are other questions and answers on here that deal with precisely this issue - saying no, getting users to go through the official channels, that sort of thing, which phrase it far better than I ever could - I'll find the links later.)
3) "I also enjoy being well respected by my colleagues, although higher management willfully ignores my performance as they do not wish for me to move to a different position"
Finally, if you're overworked and taking crying breaks, this is a big deal, and management should be aware of it, and hopefully willing to do something about it. You're obviously an asset to the company and well-liked, and they don't want you to leave. As I said, think about what your preferred outcome would be, and start to reassign some work if you can. You can even tell your manager you're unhappy, and that you're in the process of figuring out what outcome you'd like, but ask if it would be acceptable to have a meeting once you've done so to talk about your role.