Read better books? I mean, if you're looking for book recommendations, one approach is to seek out books that are written by professional historians of science attached to universities, written in the last 30-40 years. This is not to claim that all academic historians of science are non-whiggish, or that all non-academics who write about this subject are whiggish, but if you've gotten a PhD in this subject in the last three or four decades in principle you were taught how to be non-whiggish. The problem with "the history of science" as a subject of writing (as opposed to a discipline) is that it is often written by scientists (who are often unapologetically whiggish, for reasons Kuhn articulates very well) or popular writers (who are often more invested in a traditional idea of a "good story" than the complexities of historical realities).

There is a huge academic literature in the history of science, so it's hard to know where to start with recommendations without more specificity. But since you have identified physics as a particular point of interest, I might suggest Helge Kragh's Quantum Generations, which is a history of modern physics written by a very careful historian of physics. It is non-whiggish, occasionally technically detailed, and does a good job of summarizing various "big" interpretations/arguments by historians of physics, which makes it a useful place to start for that field for people who are interested in this as a serious matter. (It is too technical to be considered "general audience" — Kragh is not above using equations, for example.)

For Newton, the standard (very long) serious historian of science biography is Westfall's Never at Rest. For Einstein there is much to choose from. Galison's Einstein's Clocks, Poincaré's Maps is an attempt to be non-whiggish and non-Great Man in talking about Einstein's work and its context. Albrecht Folsing's Albert Einstein: A Biography is one that my Einstein-scholar friends recommend. For more popular writing on both Einstein and Newton that is not bad at all, Tom Levenson's books (Einstein in Berlin, Newton and the Counterfeiter) get good reviews despite being pitched at a non-academic audience (Levenson is an academic science writer, so he sort of splits the difference).

Can biographies be non-Great Man and non-whiggish? They can, but again it is rare and tricky. Janet Browne's biographies of Darwin are not Great Man, even though they use Darwin as a focal point (they are really all about the interaction with a specific person with a larger context, and the ways in which that interaction gets facilitated). Bruno Latour's The Pasteurization of France is another (more sociologically inflected) approach to this kind of thing — it's an explicitly non-Great Man, non-whiggish attempt to describe a nonetheless important individual.

Someone else mentioned this, but definitely if you want nonwhiggish science history, read "Setting Aside All Authority: Giovanni Battista Riccioli and the Science against Copernicus in the Age of Galileo" and you will like it. It goes into the science of the big argument about whether Earth moves, and when you get done with the book you will think it was Galileo's opponents who made more sense. It's not great-man-after-great-man at all.