Hi Malin,
I believe I have found your Anna Olofsdotter's birth record. She was born 1774-06-23 in Thorsberg, Ödeborg, Älvsborgs län, her parents were Olof Nilsson born 1736 and Ingrid ?Rajmayd? born 1746, Anna's birth record can be found at Födde, Vigsel, Död, 1748-1793 C:3 page 103, Ödeborg, Älvsborgs län. Unfortunately the HFL for 1771-1781 are not in the records on Genline, but the parents do appear in the HFL for 1761-1770. The family moved around 1780 to Högsäter Skatte, Färgelanda and the family can be found in the HFL 1781-1790, Högsäter Skatte, Färgelanda, AI:4 page 98.
If you have access to Genline, Anna's birth record can be seen at GID: 1018.14.34900, page 103 and the family can be found in the HFL 1781-1790, AI:4 at GID: 791.17.112100, page 98.
Hope this helps!
Have a nice day!
Don