Dry itchy eczema around my mouth

Hi I'm 16 and I have always had a problem in the winter ever since I was younger with eczema on my arms and around my mouth.

Around my my arm in the joints it's no longer there and hasn't left any scarring.
However my mouth is another story. Recently the British weather as many may know, has been terrible! It's really cold and has been during the half term, during that time my mouth began to get slightly irritated, a day later it split and cracked the skin and felt very tight, no moisturiser or oils would help nor would petroleum jelly. I tried using coconut oil but it only seemed to make the skin even more drier. Soon a couple of days later the skin began to flake and on my cupids bow the skin just peeled off in patches ... It burned and was shades of pink and red and every time I would drink water or wash my face it would burn like hell that it would make my eyes water.

Soon it just disappeared oddly when I started school again, however this weekend my skin again became tight and inflamed and randomly cracked on the cupids bow and corners of my mouth, I knew that something like the half term would happen again, little did I know that it would be any worse, my skin began to flake again to a point where I had to peel of the flakes , if I were to exfoliate it, it would make it even worse as my skin is very sensitive( I had an issue where I exfoliated my face everyday for a week and my face was all scaley and flakey like I had been burnt or something as it was patchy and darker and very sore). I stopped applying the creams that I use as they burn the skin, I tried sunscreen but it's burns and so does using normal water on my mouth or sipping on water. The scaley dryness has spread down to my chin and is closer to the ends of my nose, it feels very uncomfortable and I am in worse pain than the half term dryness. It is horrible and I need help as to what will make it any better? No Vaseline works, carmex, baby joneston lotion... It just doesn't help the itChig or dryness! Thanks!
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