Thanks for reaching out and asking a very valuable question, and I'm sorry to hear you struggle with those.
Firstly, to me dealing with something means being proactive in addressing it, and working towards its resolution. Seeing what control and responsibility you have over a situation and applying methods to actively "deal with it". Coping with something to me means it's something that it is realistically there, but potentially out of your control or current ability to deal with. So you develop coping strategies to control things you can control in order to make sure other aspects of your life are balanced, healthy and stable (which then actually puts you in a better position to then deal with it in future).
I get where you're coming from, but to me martial arts is an incredibly potent and wonderful vehicle for assisting in good mental health. I struggle with depression and anxiety, and actually quite severely at the moment. Martial arts helps on many different levels, and is to be honest one of my main therapies.
When you train, you train. You leave behind all ruminating, all problems, difficulties, and spend that time to better yourself, and also better understand yourself.
The physical nature of martial arts has incredible benefits. Not only does it get "stuck" energy moving again through the body, it affects your hormones and emotional regulation in a very positive way. Just the physical benefits alone and how it affects your mental wellbeing is worth an article haha.
On deeper levels it helps you remain grounded, centered, and stable by reminding you every time you train to be so. It connects you with here and now and gets you out of any narratives you feel stuck in.
Your practice can become a form of deeper self-enquiry... while training being aware of how you hold and carry yourself, your doubts that come up etc... and allows you to challenge and question the authority that you have given them, no one else. Your beliefs about yourself come up in training... and you can safely address them and see with awareness how they serve you or not.
MA training can even bring you to a connection with something deeper within you and life itself, and allows you to engage with the spirit behind life and yourself in a very intentional way.
When sparring, to me it doesn't stress you more, but it gives you the gradient opportunity to help deal with stress. When you push yourself and train hard it reminds you of all you are capable of, how strong you are, how you can overcome challenges. You learn patience, and then you learn patience with yourself in your own healing.
You can remember all your training and what you've been able to face, and use that as a reminder and anchor for those challenging times. You develop perseverance and resolve, and it doesn't leave you.
Not only all that... the social component is hugely beneficial. Seeing your mates and getting social interaction helps break that isolation we put ourselves in. And training hard beside your classmates and going through some challenging training is a truly bonding experience that helps you connect to others. It's a great form of support and encouragement.
Dealing with depression and stress is a multipronged approach, and martial arts is but one very valuable tool. Therapy is incredibly beneficial so I highly recommend that, but seriously, martial arts has changed my life for the better, and has become incredibly important in my healing.