Help! Audio & video don't sync after export; look perfect inside of Premiere Pro

-Summary: I need your help, please. A ~40min video I edited in Premiere Pro looks perfect (audio & video match up) when I watch the project from start to finish inside PP. Once I export it, the audio and video no longer match up.

Details:  I received an intact video as a .mov file from a friend who wished to have several short segments cut out.  It's a 40-min video of her giving a speech,and someone had already edited it a bit by adding an intro graphic and outro graphic which faded in/out.  It's got a constant frame rate.

I found the segments she disliked and removed them using the cutting tool. Each time I removed a chunk of the video, I cut audio/video at the exact same time, and then dragged the rest of the video back over to make it as seamless as possible so there were no blank spots of video.  In addition, I added a few graphics I made in PHotoshop to a separate video channel to overlay at certain times. I watched the video all the way through, especially right before/after my "cut points" (where I cut stuff out) to ensure it all synced up and flowed smoothly. It did. It was all fine. I exported it as an .mp4 H.264.

The first time I exported it, there was a short flash of red just a second after the video started, during the intro graphic the video had before I received it, and then - when the first speaker began to talk on the video, it was all out of sync. It was out of sync even before my "cut points" in the video where I'd removed stuff. I didn't understand this.  So I tried closing the project, rebooting the computer, restarting Premiere Pro, and exporting it again, and this time it was fine! Audio/video fine from start to finish all synced up, perfect for her to load to her Facebook group.

Then my friend decided she needed one more segment removed towards the end of the video. So I removed that segment, watched the video inside of PP, saw that all was synced up (including all of my previous cut points), and exported. This time, the finished video was out of sync again, starting close to the first cut point, even though it had been FINE the previous time I exported it.

I checked the original video because I read a comment saying if it has a variable frame rate, that can cause an issue, but it seems to have a fixed frame rate.

I'm an expert in Photoshop but a novice at Premiere Pro. And I'm really frustrated! If you can help, I'd appreciate it.

I'm using a Intel-based Ultrabook laptop (not a mac), Adobe Premier Pro from the monthly subscription (so it is their latest and greatest version.)  The video was given to me as a .mov file and I'm exporting it as a .mp4 using the setting H.264.  She wants to play it in her Facebook group.

Are there any subsettings I should be setting? adjusting? Is there an important setting I am not clicking/unclicking/selecting? 

When I export, here's what it says:

Output:  filename.mp4, 1920x1080 (1.0), 23.976fps, Progressive, Software Encoding, ...VBR, 1 pass, Target 10.00Mbps, Max 12.00 Mpbs, AAC, 320kbps, 48 KHz, Stereo

Source: 1920x1080(1.0), 23.976fps, Progressive, 00:41:35:03, 48000Hz, Stereo.

So I don't know if I'm doing something wrong when I cut and drag the video/audio pieces around. They're linked, so they move as a unit. But am I doing something incorrectly? Is there a "right way" or "better way" to do this? Also, am I supposed to do something else once I cut pieces of video out to ensure that the remaining parts sync up together? Could I be accidentally stretching something w/out even knowing it? I noticed the thing doesn't like to export unless the blue starter bar is right at 00:00:00, so I make sure it's there. If I do that wrong, could I be messing things up? I'm so confused.

Am I doing something wrong? (probably yes -- but I'm a novice at PP, so I won't know what it is until you help me.)

I really appreciate any help. Thanks in advance! XOXO - Jennifer