C.S.
My guess is the nose, but the ENT should be able to recognize a problem. I suggest you get a second opinion from another ENT.
My daughter had nasal polyps and her nose was always plugged and the nose would run without the ability to sniff, so she often had a runny nose. She didn't have any issues with drooling, but I can imagine if your nose is plugged, you breath through your math and as saliva builds, either he swallows or spills. It may not be comfortable to swallow if he is congested.
Try taking a look in his mouth and see if his tonsils are swollen. Shine a flash light up his nose and see if you can see anything going on in there. My strongest suggest is the second opinion.