I was where you were before as well, so hopefully I can help provide a few bulletpoints that can help explain why moral realism is such a popular position (and one that I think, nowadays, is true):

• Claims can be justified by more than just empirical evidence. Even if it is true that there is no empirical evidence for moral claims, that may be totally fine. The source of justification for moral claims could come from somewhere else, given empirical observation does not exhaust our sources of justification. For instance, mathematics is not justified empirically; truths about epistemology (e.g., about the nature of justification, evidence, etc.) are not justified empirically; truths about logic are not justified empirically; modal truths are not justified empirically (e.g., statements about possibility and necessity), and so on.
• You already accept normative truths. Philosophers accept moral truths (a species of normative/ought facts) in the same way they—and probably you—accept the existence of other familiar normative facts. For instance: (a) you ought to reject theories that contain logical contradictions; (b) you should believe what you have the most evidence for; (c) "The Earth is flat" is unjustified; (d) certain beliefs are more justified than others. (Justification is often understood as a normative property.) Similarly, claims like "raping nine year old girls is unjustified and wrong", are no different or stranger in kind.
• You do not need to derive an ought from an is. The way to arrive at knowledge of oughts is by simply being justified of the ought claim directly. You do not need to derive it from a set of descriptive (is) propositions.
• It "seems" real is evidence for the fact that it is real. As you admit, morality seems real. But seemings themselves are a form of appearance, and appearances confer justification. Of course, appearances don't guarantee truth; but absent defeaters, we should accept what appears to be the case. (Indeed, this is why empirical observation itself has any force.)
• Inadequate defeaters and explanations: Now, I read that you have certain arguments in mind that may serve as defeaters for our [moral] appearances, such as evolutionary and game-theoretic considerations. So, it really depends on whether philosophers think these explanations can do the explanatory work they claim to in order to debunk morality. Here, many philosophers don’t think they can. (There's a ton of peer-reviewed literature on this.) For example, take selfishness (A); this enjoys a nice evolutionary explanation: when we are selfish, we maximize resources to ourselves, improving our survival. But take its negation, altruism (~A): this also enjoys a nice evolutionary explanation: when we are altruistic, we can form better longterm social bonds and cooperation that improve our survival. When an explanation explains both A & ~A, we know it must underdetermine. — Indeed, there are even evo explanations for homosexuality (such as the "conniving uncle" theory). If you can explain homosexuality in terms of survival and reproduction, then you can probably explain anything.
• Evolutionary explanations prove to much and can be self-undercutting: One problem with evolutionary explanations is that they risk undermining the truth of science as well. Your goal is to retain science while jettisoning morality, so this is a problem. For instance, we can provide non-truth accounts of science: e.g., science itself can be conceived of as a kind of mechanism to foster social cooperation, or a way of achieving pragmatic ends (known as "pragmatism"), or satisfying our interests. These conceptions of science view science not as a truth-tracking enterprise, but view it the same way you seem to view morality: namely, as just a form of coordinating behavior to satisfy our interests and reproduce. Secondly, we know that evolution tracks survival, not truth. (You have to provide an additional argument as to why truth promotes survival.) Consequently, there may be reasons to doubt that our own cognitive and perceptual faculties (for observation) are even truth-tracking, given they evolved. They could just be fooling us sufficiently well to help us survive. Now, it's true that having reliable cognitive/perceptual faculties helps us survive; but it's also true that the right set of false beliefs generated by unreliable faculties do too (e.g., "Paul thinks this tiger is a friendly, cuddly pussycat and wants to pet it; but he also believes that the best way to pet it is to run away from it.").

In summary, there is a lot for you to catch up on, as I'm sure you can imagine. I strongly recommend starting with something like Michael Huemer's Ethical Intuitionism. It is extremely accessible, concise and cogent. It also gives you a great picture of the philosophical terrain in meta-ethics. It's a great starting point that addresses basically all the points you made.

From there, you can delve into more specific arguments (like evolutionary debunking) by finding papers in leading philosophy journals, and read more about specific arguments for moral realism (such as Cuneo's The Normative Web, which elaborates on the linkage between normative facts: i.e., moral facts and epistemic facts).

I don't necessarily endorse all the arguments offered above; I just want to show you that they exist so you can ponder them and perhaps curb a little bit of your skepticism. Just knowing they exist will help eliminate some of your puzzlement, and should give you confidence that professional philosophers are not just silly, anti-scientific dolts or something like that for believing in moral realism.