You are coping with a LOT. This is BIG stuff, and you need your partners support to ensure you are able to get to appointments, he needs to take care of the kids while you go, or he needs to come with you..
Alcohol addiction and depression are both very serious illnesses, he needs to start taking them seriously, and stepping up to support you with getting all the assistance you need.
When you feel terrible, have you rung lifeline? I used to, and it did help a lot. If you are feeling REALLY seriously terrible, ring an ambulance, or go to emergency.