Most likely the problem has more to do with replacing the carbon then adding the fish. The two in combination is the real issue.
Is your filter a hang on back (HOB)? The rule is to never replace filters until they are literally falling apart. The vast majority of your beneficial bacteria lives in the filter media, and when it is replaced you lose a huge portion of it.
During water changes, fill a bucket with water from your tank (not tap), swish the old filter in the water to remove debris it has collected and put the filter back in.
When it actually is getting to the point it needs to be replaced, jam a new filter in right up against it and leave it as long as possible so the old filter seeds the new with beneficial bacteria. That way it won't have as much of an impact when it is removed.
Carbon also isn't normally necessary unless you are removing meds, so you can consider just buying a sheet of filter media that you cut to size and stick in it's place to save some $.
Just keep doing pwc's with Prime to keep levels as low as possible until the tank re-stabilizes.