My daughter has one of those little pots and it works really well for her. Of course I laugh at her when she walks about the house with a little teapot-like thing dangling from her nose.
It sounds like you have a sinus infection and the gunk is running down your throat. You may want to gargle with warm salt water, but only if you're a good gargler because as you found out it will upset your stomach if you swallow it. Stay away from dairy products as they tend to clump up together and bind with the crud in your throat. Make sure you drink lots of clear fluids (no dairy) like a lot as that will thin the mucous out.
As for eating, listen to your body for the most part. Even though I don't like chocolate, when I have a migraine, I crave it. Same thing with low potassium. I think bananas are gross, but when my potassium is low they sound good.
Because of my illness and the pain levels, I'm quite used to vomiting. I probably threw up more times last year than my husband has the whole 20 some years we've been together. I often will crave carbs after a puking spell. Rice, pasta, toast (don't use a lot of seasonings or butter) sound good to me. I think it helps soak up the extra acid and bile that is in my stomach when I've evacuated everything else.
The most important things Fluids (sips if you are still nauseated, but more as you can tolerate it), Avoid dairy products and acidic foods, and listen to your body. Feel better.