None of these are really "feeling sorry for myself" songs, but they do produce a lump in the throat.
Classical - Chorus of the Hebrew Slaves from Verdi's opera Nabucco. Non-classical - Boulder To Birmingham by Emmylou Harris, and as mentioned on the question a couple above this one, Slipping Through My Fingers by Abba.
bensmum - good call on Willie's/Elvis's versions of Always On My Mind, and you've also just reminded me why I want to kill the Pet Shop Boys. ;-)