For a pre-20th century textbook: Modern Higher Algebra by George Salmon (1876) has exercises (with solutions).
If I may broaden the query from "abstract algebra" to more general "algebra", I note that Elements of Algebra by Euler (1770) has more than one hundred exercises. The exercises are discussed in The origin of the problems in Euler’s algebra.
Here is an example, from the chapter on cubic equations [source].
An another example (find an integer $x$ such that $2x^2-5$ is a cube) [source]
