I think the following references might be useful:

H. Weber (1895/1896): Lehrbuch der Algebra, 2 volumes. Vieweg, Braunschweig. If I remember correctly it includes exercises.

E. Artin (1938): Foundations of Galois Theory. New York University Lecture Notes, New York.

N. Bourbaki (1947): Éléments de Mathématique, Algèbre. Hermann, Paris.

I am not totally sure whether the last two references include exercises.

I think the following references might be useful:

H. Weber (1895/1896): Lehrbuch der Algebra, 2 volumes. Vieweg, Braunschweig. It includes examples, but it does not have any separate exercises.

E. Artin (1938): Foundations of Galois Theory. New York University Lecture Notes, New York.

N. Bourbaki (1947): Éléments de Mathématique, Algèbre. Hermann, Paris.

I am not totally sure whether the last two references include exercises.