Harmonica could work too (as would jaws harp), but just as with whistle/recorder it is a bit difficult when you have a cold.

With all non-wind instruments I am afraid knowing where to put the fingers is essential (but tis would be true for recorder/whistle too).

Can you borrow a tenor banjo, mandolin or uke to try out if it suits you? Tablature does help a beginner.

Best
Ernest