I got the answer from another forum, so I thought I'd post here in case someone needs the answer in the future:
"I think I found the problem: in the export dialog there is an Options... button. Open this and go to the Table tab - your file had "Single table" selected, which was forcing FW to try and recreate the layout with a single table, but your page layout is complex enough that you need to use Nested Tables or to allow FW to select the best table structure using 1-pixel transparent spacers. When I switched the Table "space with" option to nested or 1-pixel transparent spacer, the exported page looked correct."
Thanks again to the fireworksguruforum fo helping me out withthat.
You don't need to use nested tables or spacers if you slice correctly. Next time, try this:
Click on the rulers to pull guides over the image to mark off where you want to draw the slices. Keep in mind that the slices you draw will be creating a table, and that your table shouldn't have any overlapping rows or columns. Rows and columns should line up neatly above and below each other, and side by side, like a checkerboard. That doesn't mean that the rows and columns all have to be the same size. Just don't stagger slices, like bricks in a wall. If you do, your table will break and the image will look like it exploded.