Provided that the extension works correctly, you shouldn't need to manually delete the files. It doesn't hurt to try, just to be safe
Provided that the config.php is correct (correct db info, correct base_url, correct cookie settings), the only other thing I can suggest is double checking the settings for your server daemon, try to get them as close as possible to what they were before, taking into account whatever settings must be changed for your new server. Perhaps post notice asking users at home to clear their browser caches, just to be safe. It might add a bit of extra stress to your server while your regular visitor's browsers rebuild their caches, but that's only temporary.