Is it religiously permissible to be content with washing the area of emission with water and soap due to embarrassment from family and fear of contracting a fever?

What you did does not suffice for the obligatory ritual bath (ghusl (definition)) required due to a wet dream (ihtilam), and shyness from family members does not justify it. If you are able to heat water for bathing without harm, then it is obligatory upon you to do so; if you are unable, then perform tayammum (definition) (dry ablution). Your prayer, performed by washing only some of your limbs, is a prayer without ritual purity, and the Prophet (peace be upon him) said: "Allah does not accept a prayer without purification."