Feeling like this led eventually to my taking early retirement - I was absolutely barking by then. Don't get into that position.
Go to the GP and get signed off. You are suicidal (that's a key word for them) - you really are, that's what the walking into the river thoughts are, even though you probably think you would never do it.
Don't worry about the job. Walk away and concentrate on healing. I spent six months curled up in bed - you might not need so long.
How old are you and how long have you been teaching?
You don't have to work through what is wrong in the job right now. Give yourself at least two months when you eat, sleep, walk, take counselling, practice mindfulness and find yourself again.
Don't think of going back to work before your full-pay sick leave ends. It is teaching that brought you to this state, teaching should give you time to recover. If you do go back afterwards, have a phased return.
Get advice from your union, too.
The time to do this is now. Doctors on Monday, please. You are an emergency.
Good luck.