whose Chiuv is greater, a man finishing his Sholshim or a man who has yahrtzeit?

Thank you for your question. According to the Igrot Moshe (Yoreh Deah, 4:60) the person in shloshim has a greater chiuv then the yahrtziet, and therefore, has the preference for leading the congregation in davening as the shaliach tzibur. However, according to Harav Shlomo Zalman Auerbach (Yerushalaim Bemoadeah Ben Hamitzarim 303, 117) the yahrtziet is a greater chiuv than the shloshim. 'Lemaase', the solution is to split the davening from the amud. The mourner in shloshim (first 30 days) leads the davening until after tachanun and then the one with yahrtziet continues from Ashrey until the end. (Hilchot Aveilut Harav Yosef Tzvi Rimon, page 191). All the best!