I think I did something morally wrong during Uni course registration but can't take it back. How can I deal with this guilt?

Hi everyone, I'm a student in uni and while registering for courses in the beginning of the drop and add period, I think I mistakenly took a course spot opened for specific students. I really need to take the course to graduate this semester, but there are some circumstances that make this a gray area instead of being black and white. I simply cannot just get this wrong/mistake out of my head, but I really need to deal with it somehow.

1. Circumstances(of this event and my specific school):

- it was the drop and add period, opening at 9am, and I saw 3 remaining seats for the course for which I attempted to register and got in the course.

- a little later today, an announcement from the prof was posted, pleading for students to withdraw if they're not planning to take the course/don't need it due to a lot of demand. Then an inkling of fear that I may have immorally taken a spot meant for someone else rushed over me.

2. I had:

- failed to register during the initial opening, as the course was full.

- completely forgot to email the prof to request an additional spot (it is within their authority), and just naively believed that spots will open up during the drop and add period. By the looks of it, it is very unlikely.

----> this is completely my mistake. I shouldn't have done this if this course is as important as it is for my graduation. I didn't "deserve" the spot as it was for someone who had reached out to the professor. I didn't fulfill my responsibility.

3. (almost) confirmed suspicions:

- the 3 spots were for students that had emailed the prof beforehand. By the unspoken circumstances I think this is close to being the truth.

If this is the case, the professor knows who had taken a spot not meant for them, which would be me. I also know for sure that the professor knows me, as I've taken many classes before. I can't simply pretend this thing didn't happen, but the thing is, I really need to take the course to graduate as this is my last semester (I should have reached out, but I didn't). But at the same time, I can't imagine appearing in class knowing that I've done something morally unfair, and the professor knowing the fact. I'm so embarrassed and extremely remorseful. I'm sure many of you can relate to this feeling. I just can't brazenly act like things are normal as if this hadn't happened, and I need closure or something.

Should I:

- email the prof, explain my mistake, and apologize?

--> If I could, I really would withdraw and take another course to make things right again, especially if there was a student that couldn't get in because of me. But I can't, and that is what is killing me. What's the point of sending this email if I'm just admitting/acknowledging that I'm guilty? I did something wrong, even if it was unintentional, and I fear the email will only be used to express my guilt and that the prof will think even lower of me than if I had stayed silent.

Sorry for the rambling post. I'm clearly stressed. Small details aside, I'm just mainly asking if I should reach out to apologize and explain what had happened to the prof to clear my name (of intentional wrongdoing), also to tell them that I can't withdraw from the course, or would this make things worse than simply staying silent?

I know this may seem trivial, but at the moment I feel so guilty in many ways and this is eating me up alive, and can't move on. I have many faults in this situation and know that I'm focusing on some selfish (taking a spot for someone else) or superficial stuff (my reputation) in this post as well.

Please share your opinions on what you would do in my situation. Thanks in advance.