I consider that to be a normal bodily response to having a shower and I think that it has happened to you before and that it happens to most people all the time.
As for the BP, I bet just bet, that if you entered"shower and blood pressure into a search engine, you would find that some "links" (can't stand that word) would say that a hot shower would raise your BP, some would say that it would lower your BP and some would say that it would have no effect at all. My response is that a hot shower would do diddly sqat to your BP.
However......however..if you have any degree of diagnosed heart failure or are on any BP lowering drugs, then I would take care when having a shower, as you might feel dizzy or in fact flushed.
Your situation is due to blood vessels in your face, dilating in a normal attempt to equalise the temp. of your shower, to the outside temp. of the room........NORMAL bodily response.