No matter what moisturizer I use it leaves an oily sheen on my face?

I have very dry skin on my face (and generally all over the rest of my body as well, like my legs and arms). If I skip one single day of washing and moisturizing my face, I get flaky patches of skin all over the place. I usually use my face cleanser to clean my face, then apply moisturizer on my face while it's slightly damp. But when I wake up in the morning, my face is oil city! This has happened for literally every moisturizer I've tried and it's super frustrating. After applying moisturizer, it's like it sinks in but then leaves an oily layer on top of my skin, particularly around my t-zone. I just spent $20 on a new moisturizer to try and fix the problem, but this one does the same thing, if not worse. What am I doing wrong?