How can I better cope with stress without shutting down and beginning to ignore daily tasks?

Hello! Thank you for reaching out for help. Sorry to hear that your job is draining you and making you lose motivation due to excessive stress; it happens to the best of us.

First, it's important to remember not to try to swallow the whole elephant. Take smaller bites to avoid being overwhelmed with task completion; that's where the drain happens. Schedules (time management) with reward in between tasks helps to edify motivation. i.e. "When I'm done grading this class I'll reward myself with ___fill in the blank thing that brings you joy____."  Sometimes it's the small pleasures that make life possible.

It sounds like burnout is plaguing you so self-care is going to be very important now; even if that self-care is vegging out on the couch for a set amount of time to replenish yourself. There's no judgement in unplugging to rejuvenate.

I would advise reaching out to a therapist to help you evaluate what you're doing with your life to see if you're being fulfilled in it and what other issues may be standing in the way of a passion-fueled and joy filled life. A good therapist will help you cultivate techniques to help you separate self from profession and the demands required of such, thus aiding you in the fight against burnout. 

Teaching is an honorable profession but everything needs a healthy boundary on it. Healthy boundaries are necessay for survival. Therapy will definitely help you discover where those boundaries need to be and how to set them without alienating anyone or jeopardizing your job. 

I also recommend learning an effective breathing technique. It will work wonders on your mood. Breathing is fundamental in managing triggers. Mastering the art of breathing will open the door to effective stress management.

I also recommend creating a joy list of activities to defer to when you are feeling low. If joy doesn't move you to action after a proper rest then that's indicative that something more serious than stress is taking hold of you.

I hope this response is encouraging, informative and useful. Take care and prioritize yourself!