I did go through something like that recently. I have olive skin and dark lips, which I have now come to really appreciate (saves a fortune on lipstick). Ordinarily OP makes my lips really soft, but I had a bout of cracked lips recently. I was using sunflower oil. I had actually thought that it was because of winter and being indoors all the time, but who knows.
Switch oils and see what happens. I agree that VCO is one of the best for moisturising. I use it on my face daily and love it.