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The nose can be wet, dry, warm or cold and it means very little. I would be more concerned about her vomiting.
You first need to determine the cause for the vomiting. If your dog has not seen the vet recently for a fecal exam, I would do this first to rule out parasites as the cause.
Dogs can also have diarrhea because of intestinal obstructions or other causes. If you know that she swallowed a rawhide or other object (is she a chewer), you need to see the vet as soon as you can. Obstructions cannot be treated at home and most dogs cannot pass them on their own.
Sometimes, this is caused by a simple parasite problem or coccidia or giardia. You can find out with a simple fecal exam.
Dogs usually do well a bland diet of only boiled hamburger and white rice and it helps to settle the stomach . If the vomiting has subsided, you can start to add in her regular food.
If you supect that she is dehydrated, you can give her pedialyte or gatorade instead of water to increase her hydration.
Let me know if you still have questions.