Christians do not believe that Jesus died to balance up the sins of Adam. They believe Jesus' death was the vicarious punishment for their own sins.
Exactly, at least amongst Catholics and many of the Protestant religions, and I think Orthodox as well. Hence the mantra often heard; "He died for your sins." (ie Your sins, not Adam's or Eve's sins.)
I am not 100% certain, but I am fairly sure that the Catholics interpreted the ransom sacrifice as an atonement for our sins (not Adamic sin) well before the discovery of evolution.
in answer to the original question, I am not a believer, but if I were to ask any of my Catholic relatives (there are lots; one of them is a priest) they would all accept evolution and other sciences as facts. The only exception is there are a few who think anthropogenic climate change is a conspiratorial hoax. All the Catholics I know were taught that Genesis should not be read literally. They think that the fact that God can create clever complicated scientific principles which in turn create the world and universe, and leave no evidentiary trace, just show (to them) how amazing God is.
Interestingly, I think a lot of them would be troubled if you told them Exodus couldn't have happened either.