It might be helpful to follow a strand of British empiricism and to think about 'morality' as a social phenomenon, involving various 'sanctions' such as blame, guilt, shame, and so on. (So in that respect it is rather like law, though the sanctions there are somewhat different.) Your worry is that some moral principle you accept -- that it's wrong to cause serious suffering merely for fun, say -- has emerged only because of the evolutionary advantage conferred on groups which accept something like that principle. So it seems quite contingent which principles we come to believe -- as you imply, in different circumstances we might accept different principles. But, to pick up Alex's point, we have the capacity to stand back from our 'morality' and assess whether we have independent reason to accept its principles. In which case, if you believe there is a reason not to cause suffering for fun, you may think that this justifies the moral principle which forbids it (as it would also justify a law forbidding it).
If science (i.e. evolutionary psychology) can explain why I have the morality I do, does that mean morality is subjective? If what I believe about morality is just a product of my evolution and my upbringing, can I still expect other people to live up to my principles even though they may have had a different upbringing? What about myself? Can I still hold myself to my own standards or am I being deceived by my evolution into thinking it would be wrong to do so?