You have plastic logitech speakers and not so great headphones. What do you expect when you add another $400? Sure - you can buy for that good headphones, but what about amplifier? What about good speakers for music? I would consider aftermarket maybe and get some Yamaha amplifier + some floor stand Dali speakers. If you're lucky, it will be about $400 for older models (Concept 6 maybe + AX497). Or, if you prefer brand new, you can find some smaller speakers like Klipsch, Castle Knight, Dali or Quadrals. Or anything that people recommend here (for example). Then you'll need good amplifier and... still 400$ will be not enough.
No, save that $400 and wait for more money. For good sound it's worth to spend more, like for good GPU. Unlike GPU, speakers will last for years and there is no need to replace them every 4 years or so.
All this about listening music. For games, movies etc. is not worth spend even $200 imo. Do you really expect that gunshot or car engine will be nicer? For that your sound is probably enough.