Best Advice Re:Security for Changing E-mail Accounts

So, my gmail account was hacked again. This time the e-mail says "Hello, how are you?" It did NOT get the email addresses from my gmail address book. Apparently, it came from addresses linked to my Microsoft One Drive (formerly Skydrive) account because this morning they sent me a Security Alert telling me to change that password. I had changed my password on that gmail account, and my LastPass account, but forgot to change the password on that cloud. After running a security check on my LastPass sites, they also advised deleting a BUNCH of old addresses (especially ones from gmail) that they had on file for me. They also advised letting them generate their own passwords on the sites that I have with them.

Here's one of my dreaded questions that I dread the answer to. Should I delete my current gmail account to limit anymore damage and annoyance to the people that keep getting these hack emails? And if I do that, I obviously have to migrate all of my important e-mails I have in folders in that account. What's the best way to do that? Finally, are there any free private ONLINE e-mail accounts that are secure?

Any suggestions are welcome.

Thanks,
GPL

Thanks ICit2lol,

Being one that isn't much into security like a lot of folks, I'm not sure what you mean by "my security". I mean I run M'soft Security Ess., SuperAnti-Spyware. So, I'm not sure I understand how adding my gmail account to Windows Live makes it so much more secure. Granted, if it does, which would mean (I assume) that I wouldn't have to get another e-mail address, and/or migrate all of my old emails to a new address.

Thanks,
GPL

Thanks again, ICit2lol,

I REALLY tried to set up my gmail account in Live Mail, and I can't remember exactly, but it said it was having problems with my IMAP setup in gmail. I checked and rechecked the instructions, and everything was as they instructed. I just gave up.

I finally took the route of deleting the old gmail accounts I never use, and other accounts I don't use. I beefed up the security on my LastPass account, changed my passwords to randomized alphanumeric 12 character passwords. I use their authenticator scanner. I HATE all this crap, but I guess I don't have any choice these days.

Thanks,
GPL