I am eating lunch but I am not grossed out :) Believe me, I have seen and heard much worse things that snot :)
Sinus drainage can cause a cough. Some people get very runny nose when they eat and there is nothing there that can help. I think it some kind of reaction to the smell of food that makes people have this clear drainage. In your aunt's case, it is possible that the drainage is going in the back of her throat and is making her cough. When we cry, we also have more nasal discharge because the tear ducts are connected with the entire nose system, so yes, she can cough more if she is getting drainage to the back of her throat.
With the sinus drainage your aunt shouldn't aspirate but the snot can go into the lungs and cause bronchitis.
What you can do is use Saline nasal rinses. They get so much stuff out of the sinus, you wouldn't believe. Its just sterile water (not medicated) and it is easy to use.
But, honestly, if your aunt has not had a swallowing evaluation done, she should especially if she coughs often while eating. I think it has more to do with her swallowing than drainage but the only way to tell is to do a swallowing test. If she does have some problems with swallowing (while eating), swallowing test will help in figuring out how to position her head/neck in the best way to avoid throat irritation.