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You cannot diagnosis whitlow with a blood test – it needs to be a swab test of the lesion. If there is a blister present, it should be broken open and fluid gathered on the swab.
No, I don’t think it would be transferred to your other hand. Shedding does not occur from the finger as the skin is too thick for virus to come out without a lesion
If you have a concerning lesion, have it swab tested and find out for certain what it is.