There is possibly a small leak in the plumbing or no vapour barrier in the wall. Has it been raining and is this sink on the ground floor or second floor?
Excess moisture can build up in that cabinet space and stick to anything it can. Would you happen to have a moisture meter to check the cabinets themselves and the plaster on the wall?
Try running a dehumidifier for a few days and if that clears it up you know its a moisture issue.
Not sure why the large sink only does it, but again, there could be a few variables in everything.
If you could take pictures, it might help as well.