I think the problem is volume, but I'm not sure

When I record myself singing, I have issues with certain parts of the song that seem to be too loud or something. I uploaded a picture of what the waveform looks like. Not every part that has this wave form sounds bad, but some of them have this off sound. I have tried everything I can think of, including simply putting the laptop on the other side of the room, so it would be a bit quieter. I don’t really know how to fix it.

Yes your recording level is far too high. Reduce the recording level so that the blue waveform is no more than half the height of the track.
See here: http://manual.audacityteam.org/man/FAQ:Recording_-_How_To's#olevel