I find aloe vera very drying! I do use it everynight though instead of BP and it has the same effect without the side effects.
I use the neutrogena oil-free moisturiser
I've been using it for the past 2 years and it has never failed me! Make sure you get the one that says "COMBINATION SKIN" because the others have spf in it and that always causes me to break out. Its also a very light and gentle moisturiser, so if your skin is super dry, then maybe try some sort of oil (Caster, Rose-hip, Argon oil).
Other than that, i find this moisturiser great for oily skin! I only wash my face and use it at night though, because i find that when i use it in the morning it causes my face to become a bit oily during the day since its not that strong!
I was about to say aloe vera in reponse to the OP's question but it seems there's no one-size-fits-all product. I've even heard of people being allergic to plain water...
Then again, I don't find aloe vera drying at all. Quite the contrary, it's gentle and soothing. When applied on the skin, it has a temporary cooling effect (not as strong as mint) that makes the skin feel loved.
I'd recommend the brand CeraVe if your looking for a drugstore option (otherwise there are some good but overpriced moisturizers that you could find). Every person's skin reacts differently so I can't say for sure, but generally the top 2 brands for moisturizers/cleansers (drugstore) are Cetaphil and CeraVe, and from what I've read/heard CeraVe is less irritating then Cetaphil. I currently use the CeraVe moisturizing lotion on my chest,back and shoulders which is covered in acne, and it has not irritated my skin at all. It's also good for the face so I would check it out.