“An Obama spokesman, when asked about his previous comments on Sanders, referred to the president’s past comments that he would back whomever became the Democratic nominee.”
This is how educated people and experienced writers keep getting it wrong. You have a subordinate clause with the subject and verb “he would back,” and you know that standard English grammar demands an object to follow the verb. So you write “whomever.”
But what you did not notice is that the pronoun starts another subordinate clause of which it has to be the subject. Correctly understood, the whole clause “whoever became the Democratic nominee” is the object of the verb “would back.”
I don’t mind explaining this to you—I go over this territory a couple of dozen times a semester in my editing class, without fatigue—but you would simplify things for yourself if you just used who all the time.