Hot take: your white noise machine is making your thin-wall problem worse, not better
I live in a 1960s building in Portland where I can hear my neighbor's TV through the wall. I put a cheap white noise machine right against the shared wall for a year. I still heard everything, and my other neighbor started complaining about a weird buzzing sound coming through her closet. Turns out the machine vibrates the drywall, and the wall acts like a big speaker on the other side. I moved it to the far corner of my room, six feet from any shared wall, and put a $20 rug under it. The bass hum went away and the TV noise got way less annoying. If you have one sitting on a shelf touching the wall, try moving it across the room and see if the sound changes.