The answer to your question is "no, a new cable would not help."
You could answer this yourself in two ways,
1) If you work out the speed of an electrical field in a wire and then take the length of the difference between the cables going to one TV versus the other, you would have your answer. The speed of an electron is slow but the speed of the electrical field is quite "zippie" (close to the speed of light). The difference between the two TVs is 31' feet of cable. The distance in the wall plates are even smaller. It works out to be approximately 0.00000003 seconds difference (30 nano-seconds). You'll have to trust me that the speed of sound once the sound leaves the TV is much much much much much much much slower than this.
2) The other way you can prove this to yourself is to add another 31' of cable to the other TV as an experiment. But that seems unnecessary.
Assuming both TV's audio and video are synchronized when watching just that single TV (and you would know that by watching and listening to each TV independently), then the likely causes of your audio differences are 1) the speed of sound and 2) the delay in the different processing chips that convert HDMI from digital to analog in the different TVs. "Real time" really isn't real time with digital processing. There is always some delay and that delay can vary based on the chips used. It's exactly like asking why two computers don't process at the same speed. The HDMI converter chips are processors that read digital inputs and then output an analog video and analog audio signal that your TV's LCD or speakers can use.
Not sure you are going to find a way around this without spending more money. What you would have to do is to find a way to "slow down" the TV that is ahead. Many higher end AVRs have the capability to do that over HDMI. You could add a switch to one TV only and hope that the amount of delay is just what you need (unlikely).
Only "free" things you could try is to disable the HDMI lip-sync option in the TV but your problem is that whatever you do to one TV, the source will supply to both since any source can only send out one audio and one video stream at any time. Or you could make the speed of sound your friend and move the TVs so that the distance from each is just the right distance so that the two audio outputs are heard at the same time.
Also don't get hung up on "cheap" - you used the word 4 times in your three paragraphs. Just because an HDMI cable is "cheap" doesn't make it bad (or good). If you aren't get obvious defects in your picture, then replacing a "cheap" cable with a more expensive cable will just replace the original bits with exactly the same bits. And, again, the cable has no mechanism to hold onto the bits (as they are represented by the electrical wave). Those bits just travel freely from one end of the cable to the other.