The term democratic socialist has undergone a few different definitions through history. During FDR's time, it was no longer an anti-capitalist term perse. It was often used derisively by communists for example.
That being said you could view FDR's programs as democratic socialist, this is the type of activity that are in their wheelhouse. I wouldn't go so far as to say he was a socialist, he wasn't interested in doing away with capitalism, his programs were more a stop gap to keep american capitalism running amidst the great depression.
Are you asking three different questions, or are you equating Democratic Socialists as inherently anti-capitalistic? The question itself is a little confusing.
“The British socialist Harold Laski wrote in 1934 that ‘compared with. . .the unimaginative activity of the British government’, the New Deal was ‘. . .an exhilarating experiment’. Its various experiments were hardly unqualified successes, but a new burst of legislation in 1935 helped Roosevelt to an even greater electoral victory in 1936”
So, he was complimented by socialists at the time.
Source: Heale, M. J. (1999). Franklin D. Roosevelt: The New Deal and War. London: Routledge.
First the definition of democratic socialism I am going to use is:
During WWII centralized production was common. However, this was less due to FDR's political beliefs and more due to the necessities of total war. I think we can examine the UK, another country that was very capitalist going into the war, to see that it is normal for capitalist countries to have central planning and ownership during major wars. Thus his actions during WWII were not inherently "democratically socialist."
Now we need to examine FDR's policies during the New Deal. In terms of government (or centralized) spending as a percentage of GDP, the New Deal hovered around 10%. This was an increase from previous administrations, but still small. More importantly we need to examine central control of production.
Infrastructure was almost always centralized, but he did significantly increase it. Once again this isnt inherently socialist.
His agricultural policy was by far the most socialist of all his policies, it still gave farmers freedom to operate, but the means through which they did and the distribution of agricultural goods was very centralized. However, by this point in US history agriculture was a small portion of the US economy.
His banking reforms and the faith in the Federal Reserve are often viewed as showing his faith in capitalism. The financial system had, for the most part, failed. Instead of nationalizing it, FDR, focused on establishing regulations. Additionally, many thought that currency instability would be the death of capitalism. Lenin said "the best way to destroy the capitalist system [is] to debauch the currency" and the inflation problems in Germany helped the NSDAP come to power.) Instead of utilizing this to nationalize more of the economy, FDR went all in on the Federal Reserve and a fiat currency to stabilize things.
FDR took steps to centralize our agricultural system. However, his other major non wartime acts, such as regulation, increasing transfers (social security...), and investing in the Federal Reserve were aimed at strengthing our capitalist system. Thus although he took a couple small socialist measures, he was not socialist.
Can I ask a related, but more specific question to piggyback on this?
In his own era, which groups in the U.S. formed the electoral base that voted for FDR, and also who voted against him? Can someone describe in some more detail the political landscape of the 1930s in America?
One key thing to note is that in 1934 FDR refused to support Upton Sinclair, who was the Democratic candidate for governor in California. Sinclair was a socialist.
On the other side, the National Industrial Recovery Act and the NRA tried to implement some kind of cooperative management of the economy, with capitalists, labor and the government all cooperating. This combined some elements of Socialism and Technocracy, and the WPA did put people to work in government-run clothing factories. But in those days, the left-wing was a considerable force in America.