What you describe is "clumps" of discharge, which might be consistent with a yeast infection except that you have no other symptoms (redness or itching, for example). It's also unlikely that this is skin cells sloughing. What's most likely is that you're having a reaction to some of the ingredients of the moisturizer: I'd recommend that you try another variety and see if you have the same symptoms.
Women typically see maintenance results in four to six weeks of twice-a-week application of vaginal moisturizers. And I do recommend the use of moisturizers! Moisturizers and lubricants are the simplest steps women can take to maintain vaginal tissue and comfort during intimacy.