You need mass. 5/8″ drywall is your best friend. Lots of it. Fortunately it’s $8 a sheet.
Side note: carpet and egg cartons do not soundproof. Carpet introduces a bit of absorption which reduces reverb, and egg cartons diffuse sound waves, again for in-room acoustic benefit. Neither appreciably prevents sound from actually leaving the room, however. The carpet to quite a minor level because it’s heavy, but drywall will provide much better mass and therefore sound isolation
