We are sorry to hear about your eczema and urge you to seek medical treatment if you have not already done so. If you are in treatment, please discuss these issues with your physician to see if he or she has any advice to reduce the difficulties you have been experiencing.
In your case, we would recommend
preparing for mikveh with your feet out of the bath for most of your preparation and perhaps even with gloves (as one might use for dishwashing) on your hands for a portion of the time, to minimize soaking time. Alternatively, you could shower instead of bathing.
As long as your feet and hands are thoroughly cleaned, you are not required to soak them or to clip small pieces of hanging skin resulting from your condition prior to immersion.
Please contact us with any further questions.