I am not familiar with the exact rules of Billings, but I do practice NFP by checking my mucus on a daily basis.
First of all, it sounds like you need to speak to your instructor, maybe you have not understood the rules properly.
Secondly, you could try changing your method to Creighton or sympto-thermal. You might have better luck with these.
One of the biggest reasons many women complain of constant dry mucus is a bad diet which causes nutrient deficiencies (especially with zinc, calcium, magnesium, vitamin A and Iron). Try looking at your diet and seeing if things can be improved.
Also, some medications can affect your mucus by drying it up: antihistamines, guaifenesin (which is the active ingredient in Robitussin), some antidepressants, some herbal medicines (such as golden seal and horseradish), high doses of the acidic form of vitamin C ascorbic acid, prescription antibiotics, and, oddly enough, Clomid, a fertility drug, also affects mucus.
I highly recommend the book Taking Charge of Your Fertility by Toni Weschler and her website: ovusoft.com/
She teaches Fertility Awareness, not NFP, so the methods, such as Creighton, Billings, and sympto-thermal aren’t mentioned. Rather they just use the mucus, cervix position, and basal temps (although you don’t have to use all three). I basically use mucus signs only.
Please don’t give up, just try to talk to your instructor or try a different method. In some instances, women naturally don’t produce a lot of mucus, so instead of going by the classic mucus types, you really have to get to know your body and you should see some type of mucus. Otherwise, it might be good to see your ob/gyn, because it could be possible that you are not ovulating, or have some other medical issue.