Michael,
You could run the buffer tool with a 1 meter distance, and dissolve all option. then buffer again at -1 meter to shrink the overall extent. Of course, if the 1 meter increase does not matter for your analysis, leave them as the first result. There will be some small distortion due to the buffer out and in, but probably not too much, unless your polygons are quite convoluted.
Best Regards,
Jim