I don't think it was the frets, but I think I figured it out last night. Classical guitar requires your nails (on your right hand at least) to be longer, and normally my nails are short. Currently, I am letting my nails grow out, so right now they are not where they need to be.

I noticed that with my thumb, the wierd vibration came when the string vibrated against my skin. However, when my nail came in contact with the string, the vibration was minimized greatly. Once my nails grow out to where they need to be, I will be fine.