As to the OP's original premise, while my scenario of American Socialism is different from some of the more popular ones (as
@Hendryk says above, it seems to have happened some time around the close of OTL's WWI), I imagined anti-communism among Americans in the TL to have four main components.
The first is wealthy emigres- the Vanderbilts, the Rockefellers, the Carnegies, as well as their country cousins and localized dynasties. In my TL, these are commonly romanticized as 'the Blues' and tend to also include a lot of the Lost Generation types across Europe, who are less clearly aligned against the Revolution. They end up scattered across the Caribbean, Latin America and Europe and slowly fade out of the main pages of history. They simply aren't able or willing to do the work of restoring their wealth and power and so spend their time partying in Havana until they are displaced there, then retire quietly elsewhere.
The second is the standing military- as explained on the other place, the Reds lose most locations on the periphery of American Empire, because the military authorities there see no need to recognize the government in Washington/Chicago and the Reds have no means to actually assert power beyond the metropole. They generally turn themselves over to Entente forces post-WWI and greatly complicate later diplomatic relations between the former Entente and the now Cooperative Commonwealth. Like the Blue emigres, they slowly fade from prominence, simply because you can't maintain a military over such a period of time without access to the metropole, and my imagined scenario is largely bereft of the large scale Civil War seen in Russia (in part because that simply isn't what the US Army was geared up for or positioned for in any scenario).
The third are intellectuals- who run the gamut of outright disapproval of socialism to veiled critiques to, in some cases, attempts to infiltrate and co-opt the system (I toyed with an eccentric analogue to the Technocracy Movement being briefly popular within the IWW-SLP and leading to some concern about 'shadow bureaucracies' but haven't really seen a way to weave it into the text). Some, like Scott Nearing, are critics of the regime who are later rehabilitated. In the era where the TL is set (late 1980s), there is a rebellious undercurrent of fascination with 'outlaw capitalism' among many intellectuals which is itself running into the proliferation of wider imports in hard-currency shops (and an acknowledgement of inequality therein) as well as whatever 'openness' and 'restructuring' are bringing to American life.
Finally there is rural resistance- which runs the gamut from small family farms to local elites to the middlemen of all rural economies who form a rough analogue to the NEP-men of Soviet Russia. The IWW-SLP is most worried about rural rebellion in the early years of the Revolution and after- and this leads to a very hands-off approach for decades (to the racial element above- they create full equality under the law within the cities but turn a blind eye to what happens in the countryside, which horrifies a later generation of radical youth). They also adopt an intentional system of neglect towards rural development, based on their own reading of Kautsky and Orthodox Marxism's theories of rural transformation and their own fears of 'deproletarianization'.
The socialism of my TL has its own internal tensions- between the IWW's freewheeling syndicalism and the SLP's immortal science of Marxism-DeLeonism, between different generations, between cadres forged as allies through service in different regions, between centralization and decentralization- but the forces truly opposed to socialism are mostly dissolute, with only rural resistance being (rightfully) feared by the IWW-SLP.