Is it all the drawings or just a specific part/assembly. If it's only one rogue part/assembly then check the Customs & Configurations tab in those files - one is overriding the other...
I know this is a little old, but we ran into the same issue and found a solution, so I wanted to share for others who might have the same issue.
Are you by chance using Solidworks PDM? We recently updated to 2016 sp5 from EPDM/Solidworks 2014 and had the same issue. We found that the "DrawnBy" variable from the assembly or part file (specifically the one from configuration specific properties) was overwriting the DrawnBy variable in the drawing upon saving (this was true even if DrawnBy wasn't listed as one of the properties in the part/assembly; however, we could add it, assign a value, and that value would overwrite the drawing instead of it becoming blank). It was doing this through PDM (I'm assuming it was trying to sync the data cards or something along those lines). We fixed this by changing the block type of the DrawnBy variable to $PRP instead of $PRPSHEET, and limited the variable to file type extension slddrw instead of slddrw, sldprt, sldasm.
We hadn't changed any PDM settings when we upgraded, so I'm not sure why this changed, but hopefully this helps someone out.