You sound like you take very good care of your instrument. Very good.
But you shouldn't clean after every use. Only clean when noticeable buildup is present inside the reed area (betweeen reeds and coverplate. It will usually look like a hard white foam). At least a month should go by before you need to clean it.
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It sounds like you play a "Hohner Marine Band 1896".

The marine band has a small coat of wood sealant on the sides of the comb, this will slowly curl off the wood pieces and flake off.
..... I have not encountered this flaking on any Blues Harps and I don't buy the MB Delux.

It will not damage the playability of the instrument. I suggest using very fine sandpaper and lightly sanding the rest of the sealant off of the comb (The pieces around the 10 holes).

Also, a lot of people starting out on harmonics tend to have very wet mouths while playing. Saliva control will work itself out with time.
I used to soak my harps to the bone with my spit. This will make the comb swell past the reedplates (and split). It sucks because your lips will get bloody fast. Imagine rubbing your lips very hard and quickly on tree bark.
The solution forme was to use a razor knife to cut the wood down.... That would leave sharp edges and would slice my lips apart. Blood is very common after this. My entire top lip would be so scabbed up I couldn't move it.
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Just use very fine sandpaper and knock that stuff off.
If this is not the problem you are encountering, send me pictures via email at test.dewar.1896@gmail.com