No, it does not make God responsible! While God decreed the penalty for sin (i.e. death), He did not cause Adam to sin. Adam is fully responsible since he is the federal head of the human race, and Adam, unlike any of his progeny, was indeed completely free to choose to trust or not trust God since he was not encumbered with a sin nature or corrupt heart. UNLIKE any of us, Adam was totally free FROM sinful influences from within.
Furthermore, if you're going to blame the Doctrines of Grace for making God morally responsible for the Fall, then how does your FWT let you off the hook? God tested Adam for righteousness and he did this by sending the evil one to the planet to tempt man. Didn't God know what the outcome would be of that temptation? If so, why didn't God prevent the Fall? Wasn't he powerful enough? Wasn't he powerful enough to prevent Abimelech from violating Sarah? Since he protected Sarah against an evildoer, why didn't he protect Eve from the evil one?