In fact, FDR, the first Democratic presidential candidate to win a majority of the black vote (in his first reelection in 1936), promoted the New Deal to blacks with the slogan, "let Jesus lead me and welfare feed me."
Care to provide a cite for your claim? All I have been able to find is that it was indeed a Depression-era slogan, but one coined by blacks themselves in thanks for no longer literally starving. I came across nothing to indicate it was a party-invented slogan, let alone one used by FDR himself.