I think a good basic algebra text with nice problems is "Algebra" by Godement. Although I don't know D & F well, my impression is that the exercises are at a higher level in Godement. However, Godement stops short of Galois theory and doesn't try to be comprehensive the way D & F seems to.

At a higher level, with harder problems, is Basic Algebra I by Jacobson. Whether to recommend this at this stage sort of depends on whether you think the reason you did badly was something temporary like a lack of time or if you think you have genuine difficulties with algebra. I think it meets the definition of a "second course" though. The third chapter (of eight) is on finitely generated modules over a PID and is really great, and the fourth chapter is on Galois theory.

If you can read French, an easier follow-up to Godement would be "Cours d'algèbre" by Perrin, and there are a couple of books by Malliavin covering topics including representation theory, commutative algebra and Galois theory.