This might be because you’re using the wrong cleanser for your face. Switch cleansers. If you’re using a regular bar of soap in the shower, don’t — use an actual face cleanser. Bar soap dries your skin out in a lot of cases. If you’re using an exfoliating cleanser, it’s also not necessary, exfoliating daily is overkill and might be contributing to the problem. If you over cleanse and exfoliate, it just makes your skin go into overdrive.
Some gentle cleansers I suggest, for the shower or otherwise: Purpose, Kiel’s Gentle Foaming Facial Cleanser, Aveeno Moisturizing Facial Bar. I have a distrust for many facial cleansers at the moment and think the more boring, the better, and these are pretty basic and gentle and (shouldn’t) give you that dry feeling.
Yeah, it’s a no-no to use body moisturizer on your face. The best extremely lightweight ones that aren’t medicated that come to mind are: Kiehl’s and Eucerin (I’m not a fan of this 100%, but some swear by it).
o/ Jumping in to add something…if you’re looking for a light moisturizer, emulsions are pretty much light moisturizers in skincare, so definitely try an emulsion if you find your face becoming too oily during the day. I mention a couple that I’ve tried here.