IF you rushed to get the samples shipped, and they were damp, the results would be LOWER, not higher.  

The Lab that I have used.... if the samples were damp, the lab would indicate that in the report. 

To answer your question - how critical is it to dry the sample? - it is critical to dry the samples to stop the process of converting the nitrogen.  

If you pulled your samples one day, got them sent, and they were in the lab the next day, it may not have made a lot of difference if you dried them out or not.  If you pulled your samples one day, and it took 4-5 days - or longer - to get them into the lab, it would make a significant difference on what the results you had.  That is where drying to stop the microbial process is beneficial.