I apologize for the delay for some reason I could not get your question to open.
Trouble swallowing could be due to a perforation of the esophagus, scratches to the esophagus due to eating the stick or problems such as tonsillitis which you can rad about here http://www.vcahospitals.com/main/pet-health-information/article/animal-health/tonsillitis-in-dogs/726
It is concerning that she has not eaten in more then 24 hours. If this is something like tonsillitis then antibiotics will most likely be needed. Perforation or damage to the throat due to the twigs would be more concerning.
Twigs can also cause damage to the stomach
Normally I would say we could try some aspirin but we do not want to do that if she is not eating
as this could cause stomach upset.
What you can try is some honey or chamomile tea to help sooth her throat. Feed her soft foods. You can put some boiled hamburger or chicken
and rice in a blender and make it in to a soup that will be easier to swallow.
If she has not eaten this can also cause an increase in stomach acid making her nauseous and not wanting to eat. You can try 1/2 of a 10mg Pepcid AC wait 30 minutes and then offer the blended food.
If she continues to refuse to eat and having trouble swallowing then I would suggest having her seen by a vet as soon as possible.
I hope this helps give you some additional options to try Please reply back if you have additional questions and I will be happy to help. If you have no additional questions, please take a few minutes to provide a rating before exiting the site so I am compensated by Justanswer for assisting you.