Hello, I'm Dr Gary. I've been practicing veterinary medicine since 2007. I look forward to helping with your questions/ concerns.
There is a condition where we can see this difficulty swallowing and coughing/ spitting up food/ water- it's called megaesophagus. This is a condition where the esophagus does not contract down normally. This leads to difficulty swallowing and the signs you are seeing.
To diagnose this, you need to get x-rays of the throat and chest. We'll often have them swallow some contrast material (barium) to get a better look at the esophagus. This should give you the diagnosis.
The treatment is then with elevated feedings
and we'll use some meds (cisapride, metoclopramide) to get them to contract down the esophagus better.
I would also probably try a round of antibiotics in case of kennel cough
or an infectious process. I'll also usually use some Pepcid AC, an antacid, in case of gastric reflux. The x-rays are helpful though to evaluate fully.
I hope this helps, let me know if you have any other questions.