A lot of the data we have seems to indicate a few things.

I'm going to open by saying the South was basically still a one-party system. Most Southerners couldn't really conceptualize voting for a Republican, even if it was objectively in their best interest. A piece that provides excellent background on this is Political Reconstruction in Florida by Jerell Shofner. It's old literature, like nearly 60 years old, but it does a REALLY good job showing why the Republicans were basically discredited in the eyes of Southerners.

Basically, the argument Shofner and others in other pieces argue is that Reconstruction was a politically volatile time for Southerners. The idea of a multiracial society, something they had been absolutely TERRIFIED of for nearly 300 years (plus up to 50 if you feel like arguing Southerners were terrified of multiracial society in the early colonial period, but most of the literature really implies that isn't the case because of the early treatment of enslaved Africans), had been realized. Political violence was still common. It often targeted Black folks and often was not only for political purposes. Men were targeted for voting but also because of the perception that Black men were depraved individuals without impulse control whose life goal was to have sex with white women. In white eyes, sex between a white woman and Black man was always, under any circumstance, rape because of the connotation of white purity and Black depravity (of course, sex between a white man and Black women was always considered coercive too because Black women were deemed to be somehow coercive and powerful). Democrats explicitly or implicitly endorsed racialized violence while Republican federal officials actively attempted to stop it - notably by passing a law which allowed federal prosecution of individual citizens who had murdered a Black man on the grounds that the Black man's civil rights, guaranteed by the newfangled Fourteenth Amendment were violated when he was murdered.

So, the Democrats had a really strong background in support before the Civil War, and in the years after, their apocalyptic prophecies of multiracial society and federal intrusion were confirmed in the eyes of everyday citizens. One side allowed the South to get away with murder and preventing the rise of Black power, and the other seemed to explicitly endorse the rise of Black power.

When Reconstruction ended, Blacks felt betrayed, so although they mostly supported Republicans, they had been abandoned and most of them had their voting rights stripped away by laws like the grandfather clause, poll taxes, and literacy tests. So, White Southerners, running on advocacy of laws which explicitly allowed for white domination of politics (see: Ben Tillman), even in the era of supposed equality of men.

Progressivism and fusionist movements ultimately got absorbed by Democrats, who were able to stay competitive and ultimately moved to offer more economic relief to the poor. This brought SOME - though not many - Black folks into the fold of the party (often begroaned by whites) and shored up the base of the poor whites.

Also, and this is more directly to your point, when the Great Depression hit, Republicans were TOTALLY discredited as a party fit to rule. FDR and the New Deal (and also other liberal Democrats like Huey Long) were extremely popular and expanded the role and scope of the government. Farm policy was really popular to the mostly agrarian but industrializing South, and the expansion of the government in things like infrastructure was really popular. Democrats were also credited with a lot of gains for Black power at the time, so Black folks moved in droves to the Democratic party.

Southerners still couldn't conceptualize running or voting as anything other than Democrats, so ultimately, they stayed with that party. They ran on platforms that weren't explicitly opposed to Democratic platforms until about the 1950s, when Brown v. Board became law. Running for "Southern power" or "state's rights" or even explicitly to prevent a truly multiracial society in the case of someone like Strom Thrumond really shorted up their base in a time when Democratic race policies weren't entirely solidified. They were basically able to tiptoe a line where they said "Look, we run everything and our opposition is in shambles (the Republicans won the White House 2 out of the 9 elections and their two victories were Eisenhower, who was famous for the US victory in WW2), so we're set." It's important to note that generally, the Deep South voted for third party Dixiecrat candidates who ran on Democratic platforms + explicit endorsement of segregation during presidential elections. Southern whites were still Democrats; many Southerners believed deeply in the New Deal - in agricultural policies which had saved them from total collapse during the boll weevil infestation which decimated cotton crops and in industrialization policies which transformed the South from an agrarian society into an industrial one. Many folks didn't have electricity until the New Deal, and honestly, I'd vote for whatever candidate made it so that I could even have lights to turn on, too.

You asked if there was any one single policy that united them, and I would say no. The New Deal was a massive expansion of government which impacted everyone differently. If you want to call the New Deal a policy, I suppose you could, but I don't think that term fully encompasses it.

Sources:

Political Reconstruction in Florida by Jerell Shofner

Sex Between the Races by Martha Hodes

The Origins of Postbellum Lynchings by Michael Pfeffer

"Their Own Hotheadedness" by Ben Tillman (this was a speech delivered on the floor of the Senate, so you should be able to find it pretty easy)

Farm Policy from FDR to Eisenhower by Schapsmeier and Schapsmeier

Recommended readings:

Wilmington's Lie

The Earl Of Louisiana

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EDIT TO ADD: Sorry for cutting it off kind of short, I don't want to get too political and get in any trouble. Also, this was a great question, it's one of my favorite conversations to talk about because it stands to reason that the party was just too big of a tent that the canvas began to rip, and that's basically exactly what we see by the time of the battle over desegregation in the wake of Brown V. Board and the passage of the Civil Rights Act. I cannot recommend The Earl Of Louisiana enough for an in-depth look at how that played out in the Deep South.