Anyone know a way to do this in Audition? If not are there other software that does this?
Audition will let you do this from multitrack. Once loaded in the multitrack window you can delete at a common point across all the tracks. When you save the project and close you then should save the changes to the component files as you close those (make sure they're copies of the master until you're sure all is right!).
OTOH I think it is easier to flip your time scale to samples then just note the sample you delete at. When you do the additional files just cut at the same sample. It's a much easier workflow and less processing on the computer. If you edit with the scale on a time base it is very hard to be consistent across tracks. With the scale in samples you can be sure you're in exactly the same place.