He is not sober due to redness around his eyes and complaining about his head. He could be tipsy and pretending to be worse or he is drunk.
If he is pretending to be worse, his motive is clear.
He sees Dan sleeping at breakfast.
He sees Dan sleeping at work when it is dinner time.
Dan rubbed blook on his arm unknowingly.
During sex he said Dan is going under his skin and pushing his buttons.
It is not to stake his claim or play with Dan.
He is trying to figure out what is going on and maybe he thinks a drunk Jaekyung would be better to figure it out and Dan would open up to him than him in his normal state. He could just ask him directly. He is concern and afraid of answer.
It's ridiculous how Jaekyung is so llightweught when it comes to alcohol but if that's what it takes to mellow him down and give Dan the tenderness he deserves, ig we'll take it...