"Would Adam really have eaten from the fruit if he believed his wife was going to die that very day?"
Yes, why not? What if they thought they would die the instant they ate the fruit, as if it were poison? Eve could have told Adam, some time after eating the fruit "It's not true what God has said. We positively will not die if we eat the fruit. See! I've eaten it an hour ago and I'm still alive and furthermore, my eyes have opened. Here, try some..."
Or it could have gone something like this:
Adam: Dear wife of mine, what have you done?! Is that not the fruit of the tree that is in the middle of the garden - the very tree of which God has said: 'you must not eat from it, for in the very day of your eating from it you will positively die'?
Eve: It is even so, Adam my lord. But behold! I'm still alive. I positively will not die!
Adam: Dear wife of mine, the sun has not yet set. There is still ample time left for you to die and I will positively be bereaved. And what good would this life of mine be if it is without you? If I cannot live with you then I will die with you. Give me a bite and let us die together... why are you gazing intently at my loins and smiling?
Eve: Oh no my lord! I was just...umm....contemplating what you were saying...yeah...that's it. My eyes just happened to be pointing in that direction while I was contemplating your very touching and romantic words.
Adam: And why do you have that leaf over your loins?
Eve: Umm....it's just a....umm....decoration....yeah....it's a decoration....you know, like the peacocks have their tail to decorate them....
Adam: [TAKES FRUIT AND TAKES A BITE] It tastes great! But it doesn't feel like...whoa! Everything looks so much more clearer! Dear wife of mine, can I have a closer look at that decoration of yours? Do you mind taking it...off...so that....um....I can....um....take a closer look at you - it...I mean it...the decoration that is...I mean what else would I be talking about? Let me try it on....
Eve: [RUNNING AWAY] No! I'm not taking it off! You get away from me. Go get your own!