> Instead of saving the files with the 
> normal .zip, .cab, .exe, .dll, etc. extensions, they 
> are being saved with HTML extensions.

Look at the .html files: they are error pages, I 
presume (maybe a bad request or missing file due to 
javascripting, or filter against offline browsers) ; 
look also at the HTTrack hts-log.txt file

> Maybe the local cache has gotten corrupted and is 
> refusing to be 'updated' with the correct links?
I don't think so

> Is there any way to tell it to flush the local cache 
> links and regenerate it all from the site?
Yes, delete all files in hts-cache which begin 
with 'new' or 'old'