I agree, with Hall of Famer - REN is an amazing cleanser!
As for a cleanser, I've been using the Argan Oil alone as a moisturizer and it hasn't really been working for me. Someone else suggested to me that I mix it in with another moisturizer! I can't really tell if it's helping out my skin that much though, so maybe try sticking with just a moisturizer? I hear that Nurture Me by Ole Heriksen is super good...
I agree with all the product suggestions you have received so far! Boscia is a great line for sensitive skintypes. You may even want to try Ren, First Aid Beauty or Caudalie which have gentle, yet hydrating cleansers and moisturizers.
Boscia Soothing Cleansing Cream
Boscia Intensifying Moisture Pack
Caudalie Gentle Cleanser
Caudalie Vinosource Quenching Sorbet Creme
First Aid Beauty Face Cleanser
First Aid Beauty Ultra Repair Cream
REN Hydra Calm Cleansing Milk
REN Osmotic Infusion Ultra Moisture Day Cream
With little information, I can only somewhat guess what could be happening. Most moisturizers will not do that crack and peel effect unless the elements beforehand are somewhat abrasive (toners, larger granules in exfoliating washes) or it may not be moisturizing enough. Too much fragrance in a moisturizer over burned and raw skin can cause painful dry patches too. The first thing to try and solve the problem would most likely be to change the cleanser.
Cleanser suggestions to try:
REN - Since your profile says combination, Rosa Centifolia cleansing Gel. If that sounds too drying, then the cleansing milk
Skyn Iceland -Glacial Face Wash (at their official site), Pure Cloud Cleanser
Boscia - Soothing Cleansing Cream
Caudalie -Gentle Cleanser
More than likely once you get a good gentle cleanser that works for you, the moisturizer should not be too drying or burn the skin. If it does, check to see if it contains fragrance or menthol and look for ones that do not have either ingredient.
Try to prevent using your hands or shoulders as resting places for the chin. Also try not to rest the chin onto hot surfaces (cell phones heat up). Do not to expose the chin out into the colder elements for too long.
If you do use a toner in your skincare routine, skip out or find one that is alcohol free. That should help prevent the burning. Also do not over exfoliate the area as that will cause the skin to feel dry and raw. If the area seems a bit inflamed or starts developing questionable sores; schedule an appointment with a physician or dermatologist.
Best of luck!
I have normal/combo skin, and I'm having the same problem this year (in years past, it was my nose). Basically, what I do is keep with my regular regiment but add onto it a touch. I do some extra gentle exfoliation in the area of concern (whether it's the cheap Neutragena Wave or the more high-end Clarisonic) to get rid of any excess dead skin and apply extra moisturizer in the area. I find that it is extra helpful to apply almost three times as much moisturizer when doing my night regiment since my skin tends to soak up all the good stuff more easily (I also sleep burried under blankets so that likely affects things).
I recently switched over to the Sephora Instant Moisturizer, which has a really silky consistency and is definitely instantly moisturizing. I high recommend it.
If you want to go high-end, I really, really recommend the La Mer cream. A little goes a long way and it is super duper hydrating without ever feeling overly heavy and greasy. Of course, it is much, much more expensive than the Instant Moisturizer.
If your skin normally tends towards dry, follow your regular regiment but add argan oil in the area of concern (or totally swap out your moisturizer for argan oil until the problem goes away). This worked for my mother recently.