Wash the bands with warm soapy water, and you can try using alcohol to remove the oils on your legs to see if that helps helps. You can use thin plastic (I just save the bag I get them in) to keep the bands apart and free of lint while stored. I don't shave, and haven't had an issue with fine hairs reducing the staying power of the bands, but you may have more dense hair than I do

It could also be that they are simply too wide to stay on you, and this is what I'd bet is happening to you. They stretch a little while warm from body heat, and if it's too wide there's very little room to stretch before falling off. Generalized women's sizes drive me nuts, a large in that brand would fit me at 5'3 and I highly doubt you and I share a thigh size. So my recommendation is that you try a brand for tall women, or at least find one that advertises a thigh measurement.