Off the top of my head, some things that may help.
Mild, gentle cleanser. Inkey List Oat Cleansing Balm is lovely.
Use a gentle acid , you don't want to attack the dryness too fast or you may have an adverse reaction. I prefer Azelaic acid for my skin, I have rosacea so have to be careful. However you may find that glycolic or lactic acid work better if you have more 'normal' skin.
Hyaluronic acid under moisturiser will help to boost your skin's hydration levels. It's not a resurfacing acid, just a moisture-loving one.
When my skin is extremely dry I also use a skin oil at night, as well as moisturising etc. There are some lovely ones around, just a drop or two is all you need. Rosehip oil is particularly good (have a look in TK Maxx for Trilogy, they often have it) or also Squalane oil.
An instant fix for the eyes is eye mask patches. I have the Revolution Gold Eye Hydrogel ones. They won't solve the problem completely but they'll mark your eyes look good on the day! A MUA friend swears by them.
Good oils in your diet ie oily fish, nuts, avocado, olive oil etc.
Hope some of this is useful. You'll see I haven't recommended a different moisturiser - that's because I prefer layering skincare and don't favour changing too much at once in case of an adverse reaction. However if you do want to change it Cerave is great. I prefer the Night one for both day and night but use whatever works for you.