Before you go wasting money on new equipment, it would be worth checking out some of your settings.
In particular, is your TV doing ANY video processing at all? Motion blur reduction or any other video processing modes?
Many TVs will have a "Game Mode" or something similar that shuts off all video processing. Video processing creates delays / lag, and that split second will always get you killed while playing FPS online.
So, before investing more $, be 100% sure that your TV, Source, and anything else you can think of are not applying some processing that is introducing lag / delay and causing your current problem on the video being so delayed that the audio is too fast.
Also, if you are the DIY type, there may be some time delay circuit schematics readily available online.
Or look into something LIKE this. Disclaimer: I simply searched for "speaker delay" at PE and this came up. I know nothing about it, but the point is that there are probably units already available to solve your problem.
Behringer FEX800 Minifex 24-Bit Stereo Multi-FX Processor