If human morality were given from God, I would honestly expect it to be better, and not as subject to change as it clearly is. It's easy to see when looking back in history, all the things we (including Christians) would readily condemn and oppose, which we have totally changed our moral intuitions about today. How we raise our children, treat sexual minorities, heretics etc. Perhaps the most striking change is in how we see animals. I wouldn't be surprised if in a hundred years Christians will lament how we used to mistreat animals. Which would be a good thing, but it's kind of strange that if this great morality came from God, why it wasn't there in the first place.
The thing about things "feeling" better or more right just being a sensation we have, well that's true for all of us, regardless of why we have that sensation or where it came from. I mean, give someone a shot of oxytocin and it will make them more ready to bond with others, regardless of whether or not they are Christians. Destroy a guy's frontal lobe and he'll get problems with self-control, no matter what his religion might be. Whatever you experience isn't detached from your brain. The real question is if our brains relate to or reflect some sort of higher purpose. I don't see convincing evidence for that.
It certainly feels as if our moral intuitions point toward some sort of ultimate "ought." But other human traits like love and assigning value to things also have that quality, and we know they're not strictly true. For example, to me my children are the most valuable things in the universe, hands down. And while I couldn't escape that notion even if I tried, I can still use my reasoning to conclude that they aren't in fact the most valuable things to ever exist. I know it's not objectively true, yet for me it's still more true than anything else I know.
I'm not sure the idea of panpsychism is necessarily related to the multiverse. It's true it doesn't strictly "answer" anything, but then other physical laws don't either. Just because we find some physical law at work doesn't mean we can know exactly why it exists in the first place.
In my atheistic worldview, the whole concept of "should" is basically meaningless. There's no point in talking about how things should be, because they just are, and that's all we have to go by. But maybe you mean that there's not reason evolution should produce consciousness, and that may be true. It's a mystery why it evolved. The answer may be that it's just something that happens when a system (like a brain) gets complex enough, another suggestion is panpsychism. If there's anything to panpsychism it may be that everything has some extremely basic form of consciousness but that it takes extremely complex structures to produce anything like what we would call a mind. If panpsychism is true it's a law of nature like gravity.