I am a ger with one living parent. Since my birth parents are no longer Halachically related to me, do I stay for the Yizkor service? If so, what Yizkor do I say?

As you point out in the question you are not related Halachically to your biological parents and the laws of mourning for a parent do not apply (Yoreh Deah 374). If your parents did not undergo conversion to Judaism, you should not say Yizkor for your deceased parent (the language of the Yizkor is inappropriate to be said about someone not Jewish). Rav Moshe Feinstein (Iggrot Moshe Yoreh Deah II 130) writes that a Ger is obligated to honor his parents in the way that "Derech Eretz" - good sense and sound ethics - would require. You should find a way of showing respect to your deceased parent that would satisfy the demands of Derech Eretz. If, however, your deceased parent did convert to Judaism say the Yizkor (including in the wording of the Yizkor the qualification that you're referring to a biological parent only) since one is allowed to say Yizkor for those who are not his relatives - for example:holocaust victims.