Movement Keys Act Like They're Stuck

So I've got a problem that has me and my friends frustrated. My movement keys get stuck, whether it's a tap or a long press, and will continue to move in that direction, from a fraction of a second, to much longer, I'll estimate at least 10 seconds. Not only the standard movement keys, but the Space bar to ascend on a mount as well. The only movement key that works is autorun. This makes it impossible to play, especially group content. It starts off slow and builds up. I am on a laptop, using the intergrated keyboard. It only happens when I play FFXIV. When I type, the keyboard works normally. I don't have any problems moving around in any other game. I even have movement set to both WSAD and the arrow keys, and both sets of keys act up. We have tried quite a few things to attempt to fix it:

*Plugged in a USB keyboard. Had the same issues.
*Repatching the game. This gave me one night where I was able to move around freely. Upon restarting the computer and then the game, it was back to being problematic.
*Uninstalling and reinstalling the game.
*Making sure Sticky Keys was not on.
*Restarts after changes were made.
*Binding movement keys to one set of keys instead of two.
*Allowing Windows to turn off power to the USB keyboard to save energy; has had minor success. A few keystrokes still got stuck.

We're completely baffled by this and can't seem to find a work around for my specific case. As this only affects FFXIV, I was wondering if someone could lend me a hand.

UPDATE 1: Through a combination of setting the battery to Performance and implementing a cooling pad, I was able to play many hours yesterday without a single hitch. This all but confirms a friend's theory that the laptop was getting too hot and affecting performance in strange ways. I'm not sure if this is a lasting solution but I hope to update again after a few days with the results of a few more sessions. This laptop has some hardware limitations due to being manufactured when good computer parts were scarce, and another issue I have, where the monitor will blink in and out with white and black squares during gameplay, was also noticably absent. Fingers crossed!

UPDATE 2: Unfortunately after a few more play sessions it's become clear this is not a true fix. The first good session was the only one. However, the problem is not as severe as before implementing the battery power change and cooling mat.