If the AOL account is grey out that means you are using that account as the primary account on the phone; when you setup the phone for the first time it asks for an email address, this is the primary account and cannot be deleted without do a hard reset. To do a hard reset go to setting tap about and tap reset your phone.
I would recommend using an Outlook/Live account instead and then use the AOL as an email account only.
No need to shutdown apps the phone does great app management but if you need to and if you are running Windows Phone 8 just hit the back arrow till you get to the start screen. This means that all open apps are closed. If you are running Windows Phone 8.1 tap and hold the back arrow and you will see all the apps running and you can hit the (X) to close them.
I too am having a problem this morning with AOL email on my Verizon LG 4G phone this morning. AOL email was working fine until about 8:30 am this morning. I adjusted nothing on the phone or AOL at this point. It was telling me the login was incorrect and to try again. So, I went to AOL and logged in from my PC with no difficulty. I decided to change my AOL password to fix the problem. However, still the phone wouldn't connect to AOL and gave me the same message. Then, I deleted the AOL account and set it up again with my new password. The aol email acct set up fine but when the phone went to load my AOL emails, I got the same message again. I'm not certain is this is a Verizon system/phone issue OR an AOL issue and appreciate any help that Verizon or others can provided.
Well, I just did a chat session with Verizon and they told me that AOL has informed them that AOL is having an issue and don't know when it will be corrected. They suggested deleting the acct from my mobile device and restarting it but I already did all that after changing my AOL password first. Still having the problem. So, for now, it's hurry up and wait....
I found the following post out on a website status board. 'I am having the same problem with AOL on my iphone and ipad. It keeps indicating that my password is incorrect. I changed it and same problem. I can access mail from Chrome. Can not reach AOl by phone or chat. If I click on help and chat nothing happens.'
Apparently, major issues at AOL. I couldn't get anyone on either of their phone lines (tech and customer support).