I did find one issue in the logs:
03-11-2015 15:51:40.851 StorageServiceApi.cpp:72!LogResponse (NORMAL): GET domain-my.sharepoint.com/.../policy status:0/0x80000008 retry:57 size:0
The URL it is requesting contains the old UPN/Username/Email for Office 365. Apparently that is being saved *somewhere* on my system. It works fine when I use a different computer. Where else could it be getting the old information from?