Complex sentence: Cobb then became the first elected president, and served until old age compelled her to resign the leadership, still by their earnest desire continuing her official connection with them by acting as corresponding secretary until a short time before her death.
Split into simpler sentences: Cobb then became the first elected president, and served until old age compelled her to resign the leadership. She continued her official connection with the organization by serving as corresponding secretary until a short time before her death.