Can a convicted felon get full custody of a minor child even though he drinks all of the time?

One would certainly hope not.  The convicted felon part I could almost live with depending on the circumstances.  Maybe he was young and stupid and saw the error in his ways and really cleaned up his act and turned his life around.  But the drinking all the time is really a problem.  You can not care for a child if you are not sober.  So what need to happen is that yo need to make sure that the court is made aware of all the circumstances and that there is documentation as to the issues and that they are raised before the court.  Make sure that if visitation is granted it is supervised and that there are no over night visits.  Good luck.