Yes it could
Please read on though
It is important to know that correlation does not equal causation. That is to say that what your husband is doing might be a coincidence as to your symptoms. However both of those woods listed are known irritants that could cause allergy like assumptions. Wood dust and dust in general are known irritants that can trigger allergy symptoms. Entries in the Wood-Database, for both of those woods you mention, show the following:
Allergies/Toxicity: Although severe reactions are quite uncommon, Burma Padauk has been reported as a sensitizer. Usually most common reactions simply include eye, skin, and respiratory irritation. See the articles Wood Allergies and Toxicity and Wood Dust Safety for more information.
All fine dust is more of an issue over time with constant exposure as it can cause long term respiratory issues. It becomes very small particles when can become lodged in one's lungs. Please understand that I am not suggesting in the least that you have lung issues as a result of your current exposure. But I am cautioning him working in the house, in the way you describe, for the sake of the long term health of everyone in the house, including his own.
If dust gets everywhere then there should be a dust collection system in place to collect / contain this. This is especially the case for woods that cause irritation. You might just be reacting to saw dust and not necessarily saw dust from the exotic woods.
Your furnace should be filtering some of this dust, in an effort to stop distribution throughout the house, but that would be up to the filters you are using. There are more expensive filters that could deal with the microns of dust that are created from woodworking.
To expect on the furnace filter:
If lots of work is being done indoors I would be sure that your filter is graded for something in the area of 2-10 microns. If there is a MERV rating then higher than 9 should be good. This is based of the statistic from the Wood-Database which I linked to in other answers.
Forget about the large chips and visible sawdust: perhaps the most damaging element is the invisible fine dust (sometimes called “coarse inhalable particles” ranging from 2-10 microns).
Very related reading
What advantages does a dust collector have over a shop vac?
How should I dispose of sawdust safely?
What common woods produce the safest/most harmful dust?