What Treatment Is Suitable For Dry Skin And An Oily T-zone ?

I ve been using a certain moisturizer on my face for over a year now. Seems like when I run out of it for a few days, I skin gets a bumpy texture. By the way, my skin is dry without the moisturizer But before using this moisturizer ever, I ve never had any skin problems on my face. So when I start using this again, the bumps go away. some bumps around my mouth and nose look like pockets of oil. The rest of the bumps, even on my eyelids , just look like like raised skin, and they are on every milliliter of my skin, no smooth skin to be seen. Looks even worse in the sunlight. Someone help me.