Hey SMA, I took my first Abstract Algebra course last year and I had the exact same feelings. I could understand every line from a logical point of view, but I didn't understand it, i.e. I didn't see the underlying story. I saw that what was written down was true, but I for the life of me couldn't figure out why someone would write it in the first place, and the concepts never came to life in the course.
I struggled it for a whole semester and when I started studying for the exams I felt a bit better, but didn't feel a real understanding. But after really focussing one last time deeply on it, I suddenly, somewhere, made a snap, and suddenly I could start playing with it and I had developped a feeling for the concepts. I started feeling like I understood it :)
Don't fret! Abstract Algebra is indeed quite unlike anything you've encountered before, just keep paying attention, making exercises, thinking about how the things relate (and I didn't obsess about it or anything; I spent a lot of time on it, but not overshadowing much).