I am very sorry for both you and your dad. My mother also has Alzheimer's, and at one point was coughing and choking on her own saliva. Her coughing and choking, and aspirating in to her lungs, matched exactly what you are describing with your dad. She was unable to swallow, and thus eat.
What suddenly made a dramatic difference in her condition was starting to thoroughly floss her teeth with un-flavored dental floss, once a day. By thorough, I mean not just sticking the floss straight up and down in between her teeth, but curving the floss along the base of the tooth, at the gumline, to remove the plaque and food debris along the gumline as well. Almost immediately, her choking and coughing were dramatically reduced, and she was much more comfortable. She eventually was able to resume drinking and eating, with much less coughing afterwards.
The doctors were amazed at her improvement, they had expected her to continue getting worse. Please strongly consider flossing once a day for this choking condition.
Again, my mother did have some residual coughing even after she began flossing, although nowhere near as bad as before. I also gave her some salted butter, straight from the carton and not spread on any bread or mixed with anything, and this seemed to help her mild coughing as well. (I used Land O Lakes brand).
I am sorry that this reply was posted a few years after your original post. I just came across this forum, and post, today. Please accept my condolences for you and your dad.