Sound goes out of sync on 2nd clip of a long video

Hi there.  I know it's been a long while for you waiting on an answer.  I hope I can help.

First, understand that MTS is only a PACKET FILE format, which means it's part of a larger file that gets split for reasons related to the file storage format (fat32, ntfs_--yes I mean hard drive formats).  In order to utilize these files as a single file, you need to load the video properly.  Media encoder has had the ability to STITCH such files together since 2016 as far as I know, I believe it was much earlier.  Adobe media encoder comes with any video capable Adobe App and will fix this for you.  You should get one long file, which is easily utilized in Premiere.  Before you do this let me make one more suggestion...  When you tell it to stitch them, open the preset it uses and check the time interpolation setting near the bottom of the window.  Set it to Frame Blending.  IF you select Frame Sampling, you could have the same audio problem.  See below if you want to know why.

Now lets tackle your audio problem...

When you mention your video format, Immediately my mind is going to "Sony, Panasonic, Canon... Some kind of small camcorder".  These use a format that will DROP FRAMES THAT ARE FLAGGED as camera shake.  By DROP I mean they don't record the frame at all, just note that it's missing.  Some NLE's will try and handle this on the fly, by forcing an ingest step when you load the video, and they'll work just fine, but Premiere isn't like that.  It's what is called a FRAME ACCURATE NLE, which means it will take whatever you give it and never ask any questions.  This is done for speed of startup and access, so you start editing as soon as you open the app rather than messing with your assets (files).  So your problem then is that you are missing several frames.  Lay them all out on the timeline, they will look normal.  Play them back, and the audio falls out of sync.  This is because the background engine for audio and video only plays back each frame in succession at the specified rate; with some frames missing, those spots play back faster than the audio does, and the two streams no longer line up properly.  You cannot fix this on output.  GARBAGE IN, GARBAGE OUT.

Adobe didn't break it, but they did give you everything you need to fix it:

There are many apps I use for a project.  When I start out, I use all kinds of apps to work on scripts, event list sheets, shoot sheets.  The video tells a story right?  I have to have some idea where to start and where it ends.  I do the same in my editing steps.

I use everything adobe offers.  I use prelude, import the video, then use Media encoder to transcode with fixes for frames, and a backup file, along with audio rips, just in case.  You can probably just use Media encoder, transcode the files by allowing it to choose a setting and then set your own frame rate (uncheck the box next to it for matching the original--then set it yourself to match by numbers, without the checkbox), then check to make sure time interpolation is frame blending (best For general use).  Sampling won't work as well, though it will be faster, and optical flow is for more complex things.