You are correct about the free-will-belief and responsibility connection:

Prior to taking the math test, half the group (15 participants) were asked to read the following passage from Francis Crick’s book The Astonishing Hypothesis (Scribner):

‘You,’ your joys and your sorrows, your memories and your ambitions, your sense of personal identity and free will, are in fact no more than the behavior of a vast assembly of nerve cells and their associated molecules. Who you are is nothing but a pack of neurons … although we appear to have free will, in fact, our choices have already been predetermined for us and we cannot change that.

In contrast, the other 15 participants read a different passage from the same book, but one in which Crick makes no mention of free will. And, rather amazingly, when given the opportunity this second group of people cheated significantly less on the math test than those who read Crick’s free-will-as-illusion passage above.

There have been a number of similar studies, and the result holds up. Meanwhile, experimental philosophers have done a fair amount of work to understand what the general public understands "free will" to mean. There are some resources there that have some bearing on this thread.