That happens because your framerate dropped somewhere, Hitfilm and many would have that problem.
The video will play fine in a videoplayer but since video editors work with the video and audio separately and you can edit all that, then it just messes up.

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You can see that happens when you load the video and for example instead of saying 60fps or 30fps, it says less.
So there are only two things to do that might fix, maybe 3.

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first change the footage fps, and see if manually putting the number you recorded would fix it.
That might not, do crap... even if in some videos do, and people have used that in Hitfilm with Shadowplay.

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Second option is to use handbrake, but that would take so much, but all you need to do is make sure you have "constant frame rate" checked.

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The best way I found, and might work. is change the extension from MP4 to MOV, if when you load it on Premiere, it says the name right of the frame rate. that means it will work and be synched.
Don't ask me why, but apparently mov is read better in Premiere than mp4. some people said that even with variable fps you should be able to have your video loaded fine in premiere. I made tests and that worked for me, some people also mentioned about changing it to dif, but that apparently requieres quicktime to be installed. and I won't do that lol. but I changed my test desynched video from mp4 to mov and it worked fine.
Depending on the situation that might do the trick. but XSplit videos worked by changing from mp4 to mov, I think shadowplay was about changing footage fps inside premiere.

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Hope it works for you! if not your only solution is handbrake but that takes a lot of time to convert anything. In my test it took so long I took a break, came back and it converted and the video was synched, timer was different and changing the extension fixed without using many minutes of my life.