This is extremely frustrating, I know. The problem can sometimes be in the quality of the original source video. When I first began using Movie Edit Pro 14, I had the very same problem. My situation may be different than yours, but what I was doing was downloading various .flv format videos from internet sites, then converting them in another program, and finally importing them into MEP 14. As it was, while inside the program preview, everything was going quite well, then when I encoded and burn my movie, the audio and video were out of sync.
 
There were two reasons which I found for this:

1) the original source video was of a poor quality,

2) the converter I was using was of poor quality,

If by chance you are converting video before importing into MEP 14, this may be the problem. If so, here is the address for some really good *FREE* video converters:

http://www.snapfiles.com/freeware/

There is also other software which may be helpful at that site.

*Now* I also had a similiar problem with exporting from the MEP 14 program onto my pc hard disk rather than onto a DVD disc. I had a problem where the vid/audio were out of sync. In this case, the video in question had been through several different software programs, which had all inserted errors into the fps rate and so forth. Again, just poorly written software issues.

One thing which you may consider is trying to export to a DVD disc, rather than to your hard disc on your pc, which is what it sounds like you are trying to do. One thing which bears remembering is that when you export onto DVD disc in the burn dialogue/screen, MEP 14 will make a DVD image in the folder which you had originally designated for burns. You can then use the Mpeg 2 files in the image if you plan to use Mpeg 2 files.

Reciprically, you can then download a video converter from the above mentioned site "snapfiles" and convert the Mpeg 2 to any other format. This may not be an end-all solution, but it is the best that I can do right now. Hope this helps.