1. Don't use public computers for something secure
2. It is possible to change an email address WITHOUT being logged in.

a. go to aa.com
b. go to login
c. go to forgot password
d. fill in your name and account number
e. the next screen says

Login - Need a New Password


Verify Email Address
Is this still your correct email address? (address is partially hidden for your privacy):

**************@GMAIL.COM

f. select "No"
g. you will be asked verifying questions (pick the mileage that is closest to yours, and select you elite level, it also asks for your zip code)
h. update your email address

Note, this is far more secure the previous method. As I recall, you simply had to know the name, number, and zip code