You are correct that Bell's theorem, in conjunction with accumulated experimental results over several decades, have demonstrated to a high degree of certainty that quantum reality is non-local.
The quantum equations can be reorganised in many ways, notably done by Bohm & Hiley to describe locally real particles accompanied by a "pilot wave". But in any such reformulation, nonlocality must manifest in the nature of the pilot wave (or similar) and its interactions with the particle. Otherwise it will not be able to predict the outcomes of all those experiments, and nor will it be an equivalent quantum formulation any more but a competing physical theory.
Suggestions that the universe is therefore locally "real", just because the particles are, beg the question as to what is meant by such "reality" if the fuller reality entails the presence of nonlocal phenomena directing them.
Perhaps ironically, Bohm got philosophical and his motivation was as much to draw out the nonlocality via what he called the "implicate order" of the Universe; cutting the locally-real particles out of his pilot wave was in a sense just a by-product of his search. But even he had no concrete proposal on how the wave interacted.
You ask about the case of two [entangled] particles having opposite spin, when separated in space. The relevant properties of the space-separated pair are described by a single quantum wave equation. Any "determined-at-source" model is an example of local realism and fails Bell's test. Thus the entanglement is intrinsically nonlocal. (This was the essential demonstration of Alain Aspect's seminal experiment). But whether the subsequent measurement events may be retrocausally related remains a matter of deep debate. For example one may seek to distinguish between quantum (measurement) causality and temporal causality, allowing the apparent (classical) causal flow of events in time to be locally subjective.
Some of the issues over what "causality", "local" and "realism" mean to different people in this context are examined by Adrian Wüthrich in Locality, Causality, and Realism in the Derivation of Bell’s Inequality