I believe it goes deeper than that--and this is why the Fall is so bad:
Adam knew Eve had just sinned. I bet the thoughts in his mind at the time is that Eve was going to die--either spiritually or physically. So he chose to be with eve rather than God.
Adam consciously forsook God and abandoned Him. That was his sin.
That’s a good take on it, too.
The difficulty is that there’s limited information.
What we do know is that something about it subjugated Eve to Adam, and specifically had sin transmit from Adam.