It's partly a question of how good your ear is for sounds, not unlike an ear for music. But even though that's partly inborn or determined early in life, it can actually also be trained to some extent, even after puberty. I've told before the story of the lady in my church choir who learned to roll her R's (which I still can't do) at age 65 or 70.
I don't know whether paying for a full-fledged course would be worth it, and I'm not sure you really want an accent that sounds too American (or wherever). You want to sound like yourself, not someone else -- and it's true that Americans often find slight accents attractive.
To me the question is more this: Are there any particular sounds, or words, that native speakers find really hard to understand when you talk, or that jump out as unusually German-sounding? (Like, say, Aahnold's pronunciation of Kullyfonneeaah, which unfortunately is a word he has to say a lot?)
If so, it could indeed pay off to concentrate on those sounds. Then, even if you still have your own natural accent, it won't be as strong or as great a distraction. And if you focus on only one or two points that seem to offer the greatest room for improvement, you'll get more bang for your buck, whether literally from paying for a course or figuratively from investing your own time and effort in practice.
To do that, I think you do need a few opinions from native speakers about how you sound to them, what they notice most about your accent. If you can get those opinions, work on any suggestions they give, and then maybe even go back in a month or two and ask if they can hear that you've improved, you might not need the course.
But if the course is the fastest and easiest way to get detailed feedback, it might indeed be worth it.