AyC cigars are made with homogenized tobacco leaf. They do not remove stems before the leaves are fed in a machine that chops them into little pieces. The bits of stem are probably what you saw.
I would recommend you try a local cigar shop and look for some inexpensive cigars. You should be able to find several in the $1-2 range. While they may not taste the greatest they are far better than the AyC or Garcia Vegas you find at drug stores. They are also more likely to be properly humidified.
