My undergrad algebra was pretty much the same as yours. The first course covered groups with a focus on finite groups, the basics of ring theory including the abstract Chinese Remainder Theorem, PIDs, UFDs, and Euclidean Domains. The second course repeated the same content on ring theory using the same text, reviewed linear algebra for a week, briefly introduced modules, covered the classification theorem for finitely generated modules over a PID without proof, covered basic field theory, and did a brief survey of Galois Theory.

I can't see how treating these subjects in the reverse order would really work. Field theory requires a lot of the results about polynomial rings because extensions are constructed as quotients of polynomial rings, so I don't see how it could be taught to students who don't know the basics of ring theory. I also remember using group theory (at least Lagrange's Theorem) when studying finite fields although that can probably be avoided. Additionally if fields are taught before groups then basic Galois theory cannot be taught right after field theory is introduced, and that seems strange to me.