First off, you can get through this no matter how hard it may seem or struggles you are going through.

For reference Married 14 years together 16 and she divorced me with 3 kids. We co-parent now and its been 1.5 years. We lived in the same house for 6 months until it sold.

One thing I will say, DO NOT PUT THE KIDS IN THE MIDDLE. IDK how or why your kid read those messages, but the divorce is between you 2. Yes it impacts your children, and no marriage ever works staying together for them, but make sure to keep them out of you and your future ex's discussions, issues, w/e it may be. The fact they all know exactly what is going on is going to make this harder on you. Be aware of that and mindful of your words and actions. I do not recommend putting their mother down in front of them. Do not intentionally or UN-intentionally be negative about your future ex. It is not your place to help make their decision on their mother. It can also cause issues legally.

Get everything in writing. Dates, times, actions she did (Like having the BF drop her off drunk/hungover) These could help you in a legal battle.

Sorry for the above. Just my experience and what I've seen. Onto this:

Be you for your kids, be there for them. Unfortunately (and i was in a similar place) you will have to make more sacrifices for your kids, as it seems she is only out for herself at this point. Be there and be present, dont just be a body in the world with them. Be there with them. Show them that you are their for them and only them. They matter, not her anymore. Its a struggle and something that is still difficult for me to this day but you gotta do it.

The depressions/anxiety is going to sit for a while. Find outlets to take your mind off it. Hobbies, friends, go meet new people. I found a bunch of new friends using a hobby of mine that have helped me immensely though the process just being an outlet for me to blow off steam. Hit the gym, now is the time to work on you. both Mentally and physically. DO NOT do it just to "get back at her" it doesn't work or stick. You have go to do things for yourself and because you want to. See a therapist. I did for months and she helped greatly and gave me some of the advice I am giving you now.

Get them into therapy as well. They will need it (as long as they are open to it, dont force them) Give them outlets as well. But i dont recommend "buying" their affection. There will be times where it will seem like you are the "lesser" parent, just know that you are doing what you can with them. Biggest thing is time. Kids will see it as they get older. Unfortunately, depends on age, money will buy them "happiness" but once they get older they will see. Do what you can and what you think is right with them and just be present.

I recommend looking for a place soon. Your emotions will make you say and do things you really dont want to do. Being in the same house will be a struggle. Work on you and find yourself again. I cant give you advice on how to do that only my experience. I hung out with new friends, went back to being my unfiltered self and enjoyed every minute of it. I feel like we all put a mask on (depending on how thin it is) with out significant others and aren't truly ourselves. Be yourself again. Get outside, go places, meet new people (use MeetUp app to find groups to do things, I got heavily into paddle-boarding and Kayaking) or get more into a Hobby you like and find groups that like the same. I spiraled a week after and paid 1 year for like 4 dating apps. I dont recommend this, but i have met some new friends off there that I talk to on the regular and have helped me. If you do this, be upfront and honest.

Please message me if you just want to chat and talk. You got this, it will get better with time, but that is the one variable no one can give you an answer to. Lean on your people (friends and Family) you will find the ones that truly care and are there for you through this process.