There is no problem about changing the name by which you are known. Legally, it is sufficient simply to start using the other name (think of it: that is all that women do when they marry and choose to adopt their husband's surname). It makes good sense to notify explicitly some people and institutions with whom you have dealings -- banks, credit unions, clubs, libraries, etc.
I understand that it is a very big deal to have a birth record changed, even if it is inaccurate. I suggest that you start by contacting the registrar and asking for advice on how to proceed.