Actually, science works DESPITE many scientists not having integrity.
That's one of the reasons that I actually FAVOR creation science despite the fact that I think it is not a correct theory.
Because there are a ton of scientists out there who do fudge their data, or take their data and deliberately misinterpet it, beyond those who take their data and accidentally misinterpret it.
Then others go around and BELIEVE the studies, far in excess of the claims of the study. So then you have general academia all saying "well, science proves that lack of funding will destroy education." Or whatnot. Because they heard it, and want to believe it.
And there, to break through all the dogma, is the creation scientist who says, "uhhh.... a lot of your data looks okay, but you draw absolutely nonsense conclusions from it. Here, look at the limits of accuracy of your pb-pb dating, and consider...."
Now: Don't take my word for it... I'm making up numbers to give the gist of what I'm about to say, but 90 percent of the scientists will refuse to listen to what he said. 9% will attack him virulently. And 1% will say, "That's very interesting. I still think the theory is basically valid, but what would that imply, then?" And sometimes he discovers a fraud. Sometimes he discovers a new law.
So it is the creation scientists who battle the dogma, and help ensure that science moves forward.
So don't worry that science wouldn't move forward if scientists didn't have integrity. For it is very possible (and indeed, from what I have seen it is moderately often true) that don't, and it does.
You just have to figure out what to ignore.