When a new theory is introduced and predicts phenomena that the previous theory couldn't should the previous theory be abandoned? For example Einstein's Special Relativity. We still use Newtonian mechanics while we know that the theory is wrong. It may be correct in the limit of low speeds but this doesn't mean that the theory is correct as a whole. We have evidence that disprove the theory. I understand that Newtonian mechanics are easier for calculations in everyday applications. That is they gave the same result but their based on different theories. In other words Newton's theory provide a different explanation for the same phenomenon than Einstein's theory. Consider a scenario where two theories A and B compete about the nature of matter. Their assumptions are different. For example in theory A atoms are assumed to be "small spheres" whereas in theory B atoms are assumed to be "small cubes". Theory B hasn't been disproved but is harder to do calculations with while theory A has disproved but it is easier to do calculations with. In a certain range of experiments (as in low limit speed in Newtonian mechanics) they both agree. In that range if we use theory A isn't like we lie to ourselves if theory B is the best model we have for reality?
In the case of Newton's law vs. general relativity, physicists who do calculations with both are aware that Newton's law is but a useful and convenient approximation to the results of GR in the limit of weak fields. And yes, they are aware that Newton's law yields wrong results outside of this regime. But they are also aware that in that regime, doing derivations in GR is a waste of effort because the results are the same for both approaches to a satisfactory level of accuracy.
If you look at this from the perspective of someone following Popper (say Lakatos or Toulmin), the focus is no longer on theories being proved wrong or overturned. From a more subtle approach, science does not determine the truth of theories, or even their falsehood. It determines their relative effectiveness, which it endeavors to continually improve. A theory that has limitations and is sometimes just wrong might still be used until another one proves more effective. Most theories are incomplete or incorrect, we are only aiming to minimize those aspects.
Every science, even when dominated by a single approach, then, always contains internal contradictions, sometimes referred to as 'anomalies', the resolution of which guides its 'research programme'. Over time, it works out whether it can or cannot resolve these anomalies. If it cannot resolve them, then they will become important issues to attend to in proposing new approaches to replacing the underlying theory which will give a different 'programme' to future research. In the meantime, the current theory develops a 'protective belt' of special conditions and workarounds that allow it to be used despite its weaknesses.
A lot of results within physics are computationally intractable if you include the effects of relativity. So they are addressed with classical mechanics. And their results are then combined with predictions that depend strictly upon relativity (like the energy equation.) But this incompatible amalgamation is still better than the available alternatives: either depending only on the old theory which has already failed in this circumstance, or making predictions too complex to test. So these results are used, creating anomalies, which we then hopefully remove in the future. But even if we need several different solutions that apply under different circumstances in order to evade the contradictions we know are there, it is better than simply not saying anything useful about the difficult cases.
Consider proving that the relatively more narrative-orientated domain of chemistry, is completely reducible to the equations of physics. Do we consider chemistry to no longer be useful? No. It is a useful overlay for a domain of concern, even though rules from chemistry might not apply completely in a particle accelerator, say. Incorrect models of electron orbitals are even used, during the learning process, where they are 'correct enough'.