Okay, so what is likely happening (when you enter the IP-addressed URL for LogMeIn.com) is it is successfully getting to the LogMeIn web server, then redirecting to secure.logmein.com, which (of course, and unfortunately) is still resolving to the wrong/old/incorrect address on your Mac, then you're still going nowhere at the end of it all.
I use a 3rd party DNS service, rather than the DNS provided by my ISP, so if you can change the address(es) used for DNS on your Mac - or on your router instead if you want to change all the machines on your local LAN - to the ones offered by
http://www.opendns.com (the DNS server IP addresses, and how to configure for various devices, are linked off that page) you should then be getting exactly the same DNS lookups as I am (and that should mean the current/correct address for LogMeIn.com).
ted.h.