When you look at the properties of the exported video in the new project, what are the dimensons? Are the videos really the right size? If so, then perhaps the new sequence is the wrong size. If so, simply drag and drop the video from the project panel on to the "New Item" button at the bottom right of the panel and that will create a new sequence.
FAQ: How do I choose the right sequence settings?
Otherwise, you may have to go back and make sure you are exporting the right size and pixel aspect ratio.
