I seem to have mostly resolved this issue. I was able to add the new account by clicking on "Change IMAP Port" and putting in the value 993 since we are using an encrypted SSL connection. Thanks everyone for getting me looking in the right direction. I tried this out on my acouunt. Whenever I send emails now, I have a pull-down menu to choose who it is coming from. If I choose my primary account the receiver will just see that it came from Account A. If I choose my secondary account, the receiever will see the following in the "From" field: From Account A on behalf of Account B.
Is there anyway to get rid of the the From field to just say "From Account B" instead of "From Account A on behalf of Account B".
Thanks again for all the help.