You are doing the right thing by not allowing him to live with you again.
You make the statement that you are happier; things run smoother and the kids behave better.
His recovery is his business. I tell that to myself on a daily basis. He needs to do the work over a long period of time.
You need to work on yourself or your recovery. I am glad that you are contacting your EAP for a counselor. Good work!
Next, you may want to set up some sort of schedule for visiting the kids instead of him dropping in....etc. If he doesn't do it willingly perhaps a legal separation is in order.