It is likely that the rash you are experiencing after the shower is not really the same as heat rash. Heat rash occurs when small sweat glands in the skin become plugged due to excessive sweating and debris on the skin; it usually clears up with skin hygiene and maybe some gentle exfoliation. The rash that you get after the shower sounds more like a "dry skin" rash.
These develop because the hot water and soaps from the shower tend to strip out the skin's natural protective oils and moisture. This is worse in winter months, as the cold dry air often has the same effect on the skin. Treatment of this rash involves limiting the time of the shower and also avoiding very hot water and soaps. Upon exiting the shower, you should immediately pat dry, avoiding rubbing or friction on the skin. Then, you should immediately apply a good skin moisturizer. The best moisturizers are greasy ones, like hydrolated petrolatum. This is because they are heavy and trap in the skin moisture better than standard lotions and creams.
I suggest talking to your primary care doctor
if these symptoms continue, as additional treatment of the rash may be needed.