AOL Passwords hacked from iPhone

I have had my AOL account for close to 15 years (since it's inception really). I've had a few problems wiht my password being hacked over the years, but nothing like what I'm going through now. I rarely check my email through their website - I use my iPhone 4S.

It all started when I was going through my AOL spam folder on my iPhone looking for a legitimate email from my lawyer. When I moved that message, it opened the next message (truly spam) that contained a link (and possibly an attachment...I don't remember). I NEVER open junk email on my phone, I usually just delete it. Next thing I know, my AOL account is sending rogue emails to every address I've got stored in my phone. It all happened in a 6 hour span. I changed my password, and updated it on my phone and thought all was good. Then I received an email from a friend that looked legit so I opened it. As soon as I did it, I knew it was a mistake, so the cycle started again. I've change my password at least three times since the second time and have used a random combination of letters (mix of upper and lower case), numbers and symbols 12 characters long, so I know it's not easy to hack, I keep having this problem daily unless I remove the account from my phone - in which case, everything is fine...the problem only seems to be present when I set up my account on my phone.

It seems like there's a keylogger or virus on my iPhone 4s. I've updated my OS, I've updated my computer, I've backed up my iPhone...what else can I do? Should I replace my iPhone, or delete my AOL account permanently and change to gmail? HELP!! I'm at my wits end and my friends are tired of getting strange emails from me!