Hi jhresident82,
I'll be speaking not as a realtor on this one, but rather someone who has worked with sound and sounds studios.
The only thing that stops sound is mass or distance. Since you can't get any farther from your neighbors
the solution has to be mass.
Let me elaborate. You won't be able to sound-proof fully unless you can create an air-tight seal around your entire apartment. Your goal is to do sound reduction, basically minimizing the amount of sound you get in. Insulation unfortunately does not stop sound, it may muffle it a tiny bit, but as a whole you need a layer with mass. You already mentioned sheet rock and green glue. Yes it's one of the methods often used, it's not a bad one, but you need to pick the thickness of the sheet rock correctly or it may have the opposite effect and amplify the sound like a resonator instead of muting it. Challenge is to stop the low frequency waves from coming through and they are the longest ones, therefore more mass the better. Often even the angle of installation of the extra layer of sheetrock may be different. Key is to break down the wave then block it.
One thing to consider is what many studios, hotels and corporate buildings use for sound reduction, the product is Quiet Rock
https://www.quietrock.com/I am sure most companies or contractors familiar with soundproofing will be familiar with it. Hope this is helpful
Warmly,
Armen