I made a birthday cake with letters. I know I’m not supposed to cut the cake through the letters on Shabbat, so I sliced the cake before Shabbat, but left the cake together in the tray. Does that help that now I can take the pieces out on Shabbat, or is that still a problem of destroying a letter?

One should not cut the letters of the cake on Shabbat, since that destroys the letter and is considered erasing it (Rama O.C. 340:3). However, Rabbi Shlomo Zalman Auerbach ruled, as you did, that if you pre-slice the cake before Shabbat, it is permitted to remove a piece on Shabbat and is not considered having destroyed the letter. The reason is, that writing/erasing is only a problem if the parts are joined together, but if they are just sitting next to each other they are not considered as combining together to form a letter. Therefore, when slicing the cake before Shabbat, one should slice the piece completely, and it is not enough to just slice through the icing where the letters are written, since we need the pieces to be completely separate to consider them not joined together to form a letter.

שמירת שבת כהלכתה פרק יא סעי’ ז