Considering the age of the documents, you definitely need the assistance of someone who is well versed in preservation if intend to keep these long term.
Just a note, I am not an expert on Historical Preservation, but I have done this process myself on the advice of a local preservationist for some of my own family documents, heirlooms, and keepsakes. Your mileage may vary - no warranty expressed or implied!
It is very possible the albums themselves are the source of the odors. If the postcards themselves are not wet (I am hoping the use of "damp" does not mean water damaged?) you could remove the postcards and place them into individual zip-lock bags and then freeze them for 72 hours. This will help to kill any active mold that may be present and causing odors.
After 72 hours, remove the postcards from the freezer and immediately remove them from the zip-lock bags (leaving them in the bags straight out of the freezer will attract moisture). Lightly brush each one with a natural bristle brush to remove any possible traces of mold. Once they are cleaned, you can then place them into archival sleeves (avoid your typical PVC sleeves at all costs).