Per you SWH, I think you need to talk with a professional, a counselor or therapist, to make a plan for yourself. You say you don't know why you stay with him and wish he would leave. That kind of thinking is very passive, and if you continue on this path, five years from now, you'll still be in the same situation. Wishing won't change things. YOU have to change some things.
You've got children watching you and how you react to this situation. You're enabling your husband by allowing him to do whatever he pleases while having no responsibilities whatsoever toward the family. It doesn't matter that your children are not biologically his, when he married you, your children became part of his responsibility, too.
Can't say whether your husband has a video game addiction, depression (or both), or is just lazy and irresponsible, BUT you can't allow your kids to think this is normal and okay for a grown man. What message does that send to them, not to mention what this is doing to you?
Even IF he is going to school on-line, a grown person with a family should still be working and contributing around the house.
Go talk to someone and figure out how to make the changes you need to have the life you deserve.
Good luck and stay strong.