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Your original expert has opted out, and likely other experts are unaware of this as your question shows it has a response. Sorry you have had to wait. Your question is a perplexing one, many times skin rashes are difficult to diagnose even by a doctor that sees you in person.
Your symptoms of red, burning, itching palms and feet has a name: "palmoplantar pruritis". There are several conditions that can cause this. Contact dermatitis (or, an allergy to something both your feet and hands come into contact with, perhaps something you are showering with for example), or a condition that is similar to eczema. Other possibilities would include food allergies, or an actual infection such as a fungal rash. Also possible is a bacterial infection, or certain systemic or skin diseases.
Here is a link to one possibility: Dyshidrotic eczema
You really do need this looked at by your doctor, and probably by a Dermatologist. In the meantime, you can try some general measures to try to quiet down the itchy, red rash response. Cleanse gently with warm (not hot) water only. If you do use soap, use a moisturizing, hypoallergenic lotion type. Use an emollient cream to keep the skin moist, as dry skin can also flare like this. Apply a hydrocortisone cream or ointment up to 4 times a day. All of this you can get at your local pharmacy.
Also, for now if you do any cleaning with soaps, detergents, chemicals....make sure to wear gloves, or allow someone else to do it until your skin heals.