Habesha have Yemeni ancestry from the bronze age, thousands of years before Axum conquered Yemen
The African and Near East Bronze Age ended earlier than the European, so technically, it would be Near Eastern early Iron Age. A study claimed the coalescence time of the earliest single introduction of the South Arabian language to be 2800 years old in modern Eritrea and Northern Ethiopia. Basically, Agaw-like people, language shifting and mixing with Yemenite merchants.
Sources:
The evolution of languages provides a unique opportunity to study human population history. The origin of Semitic and the nature of dispersals by Semitic-speaking populations are of great importance to our understanding of the ancient history of the ...
www.ncbi.nlm.nih.gov
Possible good explanation of how it likely went down (page131):
Christopher Ehret, the author of the section in the second link and is also a co-author in the first link that calculated the age of Ethioseimitc. The second explains the possible language-shift process.