dry red and burning sounds like something a doctor should diagnose.
however I'll recommend a few things to you concerning your eczema and cold sore.
For the cold sore, taking L-lysine usually helps people. At the start of an outbreak take 4-12 tablets a day or ask a doctor about the correct dose. Taking a small amount regularly may also reduce the number of outbreaks, do some research on that. Lysine is produced by your body naturally.
For the eczema and dandruff, I'll recommend you research products made by Salcura. Salcura is a British made product line so you'll have to order it over the internet. Salcura deramspray gentle is what we used on our child who had severe eczema. It was 10 times more effective then everything else we tried. It stopped the itch in seconds and healed the inflamed red skin rapidly as well. It's all natural, over the counter, leaves no visible residue and is easy to apply Salcura also has products for the hair and dandruff. Psoriasis sufferers have reviewed their shampoo and said it was the best they found for their condition. Also Salcura Antiac is the best product for acne according to a British tv show named "Embarrassing Bodies" It beat out all the other products they tested. You can find the video on you tube or at expresschemist:co:uk.
Something in the cleansing product you used may have a trigger for your eczema, so yes it's possible the cleansing product is something you should stay away from. Do a small test on your face or skin before using products.
hope this helps you.