The stuff you threw-up could have come from your throat or stomach, especially what you swallowed the 2 to 3 times before this chicken lodged in your throat. The real question is did some part of the chicken remain in your windpipe. If so, especially with worsening pain, you better have a chest x-ray. If anything remains in your windpipe, it may have to be removed.
Severe, sudden vomiting can lead to a small tear or rupture in your esophagus. You may have expelled some fluid from your lungs from the wretching?, but keep a close eye on the level of pain you are experiencing. If it becomes worse, I would go to the Doc/ER for an exam to make sure you haven't injured something in the esophagus.
I've had problems for years with a constant cough, but the past few years has developed into a really hard cough when I eat. I often get strangled on food, cough til' I pass out or throw up the food and some mucus. After some tests, the dr put me on Bethanechol Chloride. It works on the esophagus to make it move the food down. Have been on it about a week now and have not gotten strangled from eating at all. Maybe this will help you.