Science does not undermine its search for answers at all. We know this because of its very nature of asking questions and always finding new ones to answer. In fact, one may say science propagates itself, and even expands exponentially, this way.
To see this in action, it help to take a brief look at the process of science. Science deals with making observations (scientific experiments) and forming hypotheses (educated guesses) to explain those observations. Scientists repeat this process to refine their hypotheses until they predict the observations as accurately as possible. Now it would be nice (and a bit boring) if they found found a perfect hypothesis that exactly fit observations the first time, ending any further inquiry. But this has never happened. The reason for this is that careful observation of nature at the boundaries of our knowledge usually reveal exciting new unexplained phenomena. The old saying is very true: "Answering one question usually uncovers many more unanswered ones." We can see how searching for answers tends to build on itself instead of undermining itself.
One may think that eventually after a lot of searching and answering we would arrive at a "theory of everything" (which scientists are currently pursuing), and that would lead to an undermining of science searching for answers. We have clear evidence that even if we had a "theory of everything", science would still be searching for answers. For example, we already knew the science of electromagnetism to a high level of accuracy by the mid-sixties but that did not mean we knew all the answers to develop today's electronic technologies. We know the basic principles that apply but we do not yet how they apply in specific situations. This should tell us that while we may know the fundamental principles very well, we are far from knowing all the possibilities of those principles. This applies equally to all the other sciences. It should be abundantly clear that science is far far away from becoming obsolete.
It may come as a disappointment to some that all the phenomenon in the universe will one day be described by a small set of physical principles. I can only offer the following perspective as consolation: While we may see the Big Picture, we have yet to see its intricate details. Those infinite details are the answers we have yet to find, and science will forever be searching for those answers.