SQL Server MDF still has old values after update
This is probably one of the silliest processes I've ever heard of, but anyway.

After you've updated your data to remove the offensive words, run a checkpoint, and then restarted SQL Server (just to make sure it's all committed), the only way to guarantee that the underlying disk has been overwritten, and the "offensive" data expunged, is to perform a shrink of the database file.

This effectively takes the data pages from the end of the file and inserts them into open spaces at the beginning of the file to fill up "empty" space. It's an horrific process for SQL Server, as it causes massive fragmentation, but it will "solve" your problem.

Once you've moved your data to the new server, you must run a full index rebuild to defragment the tables, but, hey, it should work.