hi Leandra,
i searched google and found only these 2 OBGY clinic in Tainan. You probably could start calling them to see if they have any baby born at your b-day, then find your birth mom. I think the street names should stay the same as 1981, so it ought to be easier to locate her. Problem may be 1. to get clinc releases info to you, 2. to get in touch with her. Tainan is a small city, it is hundreds of years old but not as big as Taipei.
葉能貴婦產科診所/台南市東區崇明路91號/(06)-2696947
葉婦產科診所/台南市中西區忠義路二段53號/(06)-2220265