I just open a second IE window for my mule. Get logged in as one character & go to a quiet place to wait. Then open the 2nd window, log in as the other character & go find the 1st one. Have them "Invite to be friend" so you can find each other quickly later. Then you can just trade between them.
You can also find someone you trust here on the DIS who will agree to "hold" things from your mule while you log out & back in as your main character. I don't like to do that because then I have to wait for someone - and I'm impatient. LOL