There are two possibilities that come to mind. 1) You are having an allergic reaction to the wood. 2) You are having an allergic reaction to some harsh chemical that the thrift shop people may have used to clean the recorder.
It can be difficult to determine the cause when you have more than one potential cause. YOu may have to visits a doctor specializing in allergies and get tested for reactions to various woods used in recorder building–maple, pearwood, boxwood, rosewood, grenadilla, and cedar (used for the block).
The wood in the second picture looks like it is not from the rosewood or palisander family which can sometimes cause allergic reactions in people. It may be pearwood which is commonly used in less expensive instruments. I’m not familiar with fruit woods causing allergic reactions. However, the wood in the first picture looks discolored–possibly as if it has been treated with something not intended for use on musical instruments.
If the first case is true and you do have a wood allergy that can’t be avoided, then get yourself a plastic instrument–there are several good ones available at reputable shops (not thrift stores). If the second is true, throw the instrument away and try again with an instrument from a reputable shop.
I hope you recover and can enjoy playing.