@milktea13 There is no law in the Philippines requiring women to change their names after marriage. It's just common practice but you can do away with it if that's your choice. You can carry on using your maiden name if you want to, that's the name your parents gave you.
As for immigration purposes, it is strategic that you use your maiden name even after marriage or even if you really want to use your husband's surname if it is time-consuming and costly to have replacement documents.
The birth certificate DOES NOT show civil status... it's a report of your birth. So don't worry about it showing as single because it doesn't.
To show married status, just submit your marriage certificate and an updated NBI that shows married. Note that NBI and immigration authorities would want to have a PSA certified copy of your marriage certificate... for some instances this could take a while as well.