
It's worth actually training him to accept your going is not going to be permanent
This can be done by teaching him an out of sight exercise.
I would have a special toy like a kong filled ith allsorts of treats & this he only gets when you are going out. Give him the toy with a command like "see you later" & leave him say in the kitchen & close the door between you for 30 seconds. If he has made no noise, return take the toy & give him a tasty treat & allow him to stay with you. Repeat this at various intervals during the day gradually increasing the time he is on his own.
I would also do some real obedience training, including stays & recall.
Eventually he will relate getting the toy with you not being there, & also that you will return & he will get more treats
He may never be really happy at being left, but he will accept it in return for rewards