HerHusband wrote:

... the air gets very dry when we use it.

Not necessarily, with a fairly airtight woodstove in a fairly airtight house.
It might only need 10 cfm of combustion air, vs an average US house that
naturally leaks 225 cfm or ASHRAE's 15 cfm per occupant fresh air standard.

I'd like to buy a steamer to set on the stove...

Evaporating water takes heat energy, ie more firewood. Airsealing the house
will raise the humidity and decrease the need for heating energy, ie firewood.

Nick