Am very sorry to read of your problem, have had a woodburner for 26 years as when we moved here there was no natural gas and LPG considered to wet/smelly/dangerous.
I have asthma and COPD but have never experienced any problems like you describe; we use well dried wood stored in airy wood sheds and always have ALL doors open throughout the house [except outer doors of course] which causes good air flow.
Are you using dry weathered wood that is not allowed to get wet in storage?