Genesis 3:6 6So when the woman saw that the tree was good for food, and that it was a delight to the eyes, and that the tree was to be desired to make one wise, she took of its fruit and ate, and she also gave some to her husband who was with her, and he ate.
Is there something about about the words "her husband who was WITH her" that is too hard for you to understand?
I do not necessarily understand it to mean right next to her, or "right there" as you said. He could have been 30 feet away, picking berries or something, and not heard what transpired.
Eve could have tasted the fruit and then turned to Adam to get his attention, saying, "here Adam, eat, this fruit is good."
If only Adam is responsible for the fall, than only men would need to be saved, which is not the case. In God's eyes, both are equally guilty, but Eve committed sin first, and her punishment is also harsher then man's.