Two suggestions. First, call back and try again. Second, take to Twitter and seek some help from @DeltaAssist.
Of course, getting a refund might not be ideal. Appears the only other nonstop is AA and it's an evening flight. So you could have DL move you to their evening flight. Or, have them switch you to a connection (I travel with 3 of that age range myself frequently so I know that isn't ideal). There's a 9am via MSP, arriving at 1:15pm, with an hour in MSP, so tight with three kids, but gives you more morning time and gets you into LAX at a reasonable time. I suspect they'll be much more accommodating moving you to another DL flight versus giving the refund.