I am very sorry to hear that your fellow Dillon is drooling excessively and coughing today.
I suspect his cough is more of a gag due to excessive drool, or it may be related to nausea and acid reflux.
Drooling can be a sign of nausea but it can also be a sign of oral pain due to an infected tooth or a foreign body caught between his teeth, on the roof of his mouth or in the back of his throat. Sometimes it is a sign of an oral tumor. If he were an older fellow nausea due to organ failure would be a more likely possibility, but he is a bit young for that unless he was exposed to a toxin.
It can also be due to a nerve problem that is making it difficult for him to control his tongue or close his mouth properly. If he is eating and drinking without any trouble I believe that is less likely.
Finally it can also be a response to eating a bitter bug or plant piece. If it comes along with facial swelling or hives it could be related to an allergic reaction.
I do recommend rinsing out his mouth with cool water several times.
I know that may have you looked in his mouth but sometimes it takes a very close look under sedation to find the problem. If he will let you examine the inside his mouth again closely for swelling, reddened areas or any sign or trauma or a foreign body.
You can give him acid reducers to try and settle his stomach in case this is related to nausea. Either:
1) Pepcid-ac (famotidine) at a dose of one 10mg tablet per 20 to 40 pounds of body weight every 12 hours.
2) Prilosec (omeprazole) at a dose of one half of a 20mg tablet per 20 to 40 pounds of body weight every 24 hours.
These will reduce stomach acid and should help settle his stomach. These are quite safe and can be used for several days if necessary.
A couple hours after one of the acid reducers is given you can offer small amounts of water or ice cubes to lick.
If that goes well then late today or tomorrow start a bland diet of 1/3 boiled, lean hamburger (or boiled, white, skinless chicken) and 2/3 plain, boiled, white rice. Give small meals several times a day. Feed the bland diet for several days, then start mixing in his regular diet and slowly convert him back over a period of 5 to 7 days.
If he attempting to vomit but unable to do so, his belly looks at all distended or he won't lay down and settle or he is pawing at his mouth or unable to drink water normally then I recommend he see a veterinarian immediately as this may indicate something stuck in his mouth or throat.
If his drooling continues in spite of using an acid reducer and a bland diet then he should see his veterinarian for an examination, some blood tests to look for underlying metabolic disease, possibly sedation to look closely in his mouth and take dental radiographs.
Please let me know if you have any further questions.