"He’s already/just done the shopping, now he’s helping his wife with dinner. "
"already" gives a more argumentative feel, opposing the idea that something might not have been done. "just" gives more a sense of a progress update (unless heavily voiced).
"just" means it's done only a moment ago, "already" could have been done at any time; thu
"He’s already done the shopping, now he’s helping his wife with dinner. " (The shopping, one of his chores that was on the list, was finished sometime earlier.) "He’s just done the shopping, now he’s helping his wife with dinner. " (The shopping was finished a minute ago. He is a busy person.)