Agree with this. I don't think you've said how long it's been since you separated OP, but it sounds like you're still in the raw early stages, when you're doing nothing but crying, and not eating or sleeping. I remember that panicky feeling of not knowing when or if it would stop. Also, not being able to stop myself bursting into tears when speaking to a client, which added to the feeling that I was incapable of working.
The good news is this intense stage DOES end. In my case, it did a screeching u-turn, when after a couple of weeks ex came to get his stuff and I noticed he'd taken his wedding ring off. My grief turned to fury - an emotion very much more useful to me!! I'd like to think that everyone in this situation has that 'blinding clarity' moment that jolts them out of the pit of despair and brings them back to something like their normal selves.
Anyway this is just to say, if possible please don't burn your bridges with work. Regular income, routine and familiar faces will all help you keep it together while you ease into this new chapter in your life.
Summon up the strength to have an honest conversation with your line manager about how your situation right now is affecting your ability to do the job. You've posted that they've started picking you up on things anyway, so they have noticed. Did they know about your personal circumstances when they talked about "going down the disciplinary route"?
It's hard, but try to see things from their point of view. Personally I wouldn't go in demanding sick leave - I'd make it more about you and the company finding a way forward that you can both agree on. And as with any negotiation, it's good to have an idea of what kind of outcome you want, eg a period of paid/unpaid leave. If you're reasonable, hopefully they will be too. If they're not - you're better off out of there, but preferably at a time of your own choosing!