Yes, wood stoves dry your air out big time. They sell steamer pots to put on your stove for passive humidity, or you can run a humidifier while running the stove.

I use a steamer pot I bought from LLBean. Made of cast iron , great for heating the water but it does rust some. Best to get one that is lined with tin or copper. During the cold weather it evaporates 1+ gallons a day and the humidity in my house is around 50. When my house was built and before they put the drywall up, I lined the inside of my house with 6 mil plastic over lapped 16 on center. It helps a great deal with keeping the air less dry. I have seen other homes that painted the inside of their homes with a latex sealing type paint that provides a moisture barrier and they too have had success.

I forgot to add. For $2, a bottle of saline nose spray does a lot of good to alleviate dry sinus.