You shouldn't necessarily stop believing something because all of the evidence shows your belief to be false.
Yeah, you pretty much should.
There is something to be said for intuition.
No, there's nothing to be said for it. Intuition isn't a magic sixth sense, it's how cognitive biases in your mind manifest. They make you "feel" like some things are true regardless of whether they are or not.
If we stopped believing something because all of the evidence shows the belief to be false, then there would be times when not all the evidence was available and when that evidence did become available, our belief would turn out to be vindicated.
So? The point of knowledge isn't to be right every single time, or to get ahead of the evidence so that you can turn out to have been right before everybody else. It's not to be "loyal" to ideas and not betray them - they're ideas, they don't care whether we're fair-weather friends or not. The point of knowledge and evidence is to hold
the most defensible position at any point in time, based on what is known
at that point in time. When the state of knowledge changes, you should change with it.
Now, maybe you're thinking "but, if I did it that way, I'd be wrong a lot! I'd have to change my mind about stuff when I learned new things, and I might even have to admit I was wrong!" Well, yes. But what's wrong with doing any of that? Especially when you stand to gain so much in terms of an accurate understanding of what's going on around you?
Knowing and understanding how things work is
incredibly powerful. It gives you an unparalleled ability to influence other people, to change the world around you, to solve problems and reduce suffering - or to cause it, if that's your deal. I would prefer people use their knowledge for the betterment of those around them, but the truth is, the power of being a fair-weather friend to ideas is vast regardless of whether it's used for good or ill. You should ask yourself why you would want to close yourself off to that power for nothing more than loyalty to something - an idea - that can't know whether you ever were, or not.