I am sorry that you are stuck with this infantile mighty mouth, who seems to believe that going to AA entitles him to rule the barnyard. It may seem silly to have a dust-up over how full/empty the dishwasher was, but where an alcoholic is concerned NOTHING is too silly to fight over.
If these are his actions whilst supposedly in a recovery program, then he sure is way off track and heading for a big fall.
If he isn't contributing towards the house expenses, a good chat along the lines suggested by Anvil would be well in order. If he carries on about doing his bit in providing towards household expenses and such, then it is time to evaluate what you want instead of handling whatever he doles out.
Why should it all be on your shoulders? Maybe you could take some leave and tell him you are taking time out to "WORK YOUR PROGRAM", so he can meet the bills, do his own meals, washing etc til you feel better.
As I loudly informed my ABF when he drove me to almost murder, "being in AA doesn't make you a sacred idol....it just means you're an Alco working a program instead of a programmed alco."
I demanded equality in time and house work, expenses and such for me with Alanon as he had with his D & A counsellors, Rehab meetings and so on. It worked fine for a good while.
Mind you if this is your "sober step working" man at work, hmmmm maybe sobriety is not going to be so sweet. You know some of the sayings about taking booze out of the cattle thief and what do you have?
Hope he grows up fast and soon, and helps as he should. If not then I wish you strength, lots of ideas and enough anger to give you real motivation in moving out of his way and going your own.