Hi Olivia,
Often you need two doses of dewormer to kill off the eggs as well as adult worms. Since your pup will be due for routine vaccines with a vet, I’d advise calling to make an appointment for an exam. Take a fecal sample with you so they can find out what type of worms your pup has, and provide the correct medication. Good luck with your new pup!!
~kelly
It could be allergies or maybe something like a foreign body in the nose. I’d be tempted to run to the vet if this goes on for more than a day.