For a pre-20th century textbook: <A HREF="https://archive.org/details/3rdedlessonintro00salmuoft">Modern Higher Algebra</A> by George Salmon (1876) has exercises (with solutions).

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If I may broaden the query from "abstract algebra" to more general "algebra", I note that <A HREF="https://en.wikipedia.org/wiki/Elements_of_Algebra">Elements of Algebra</A> by Euler (1770) has more than one hundred exercises, and may well be the oldest textbook on this topic. The exercises are discussed in <A HREF="http://logica.ugent.be/albrecht/thesis/EulerProblems.ps">The origin of the problems in Euler’s algebra.</A>

Here is an example, from the chapter on cubic equations
[<A HREF="https://archive.org/details/elementsofalgebr00eule/page/270/mode/2up">source</A>].

<IMG SRC="https://ilorentz.org/beenakker/MO/Euler_1770.png"/>