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Dont worry about using a hot towel. You're not at a barber shop, where you can't shower first. Soak your face at every opportunity in the shower, then shave immediately after. Why are you lathering...

I have sensitive skin, too, and had the problem you're having, but on my neck. Pressure and a "dull" razor are the two most likely culprits. It definitely takes some time to get the hang of pressure....