Addressing the thread title, no Adam did not have a free will after the fall. Before the fall, Adam's will was not as limited as it was after the fall. But, even before the fall, Adam operated within the purview allowed by God, so it was not totally free.
Turning to the posted question, the corrupting influence was Adam's ability to choose to go against God's will. Without that, he would not have chosen to sin and eat of the tree of the knowledge of good and evil.
OTOH, God had chosen Christ to be His Redeemer, His "Lamb of God" before creation (i.e. before Adam was created) and so God certainly anticipated Adam's sin and the fall of humanity. Thus the forbidden tree fruit and the allowance of Satan to deceive Eve. We cannot say God compelled Adam to sin, but God certainly provided the opportunity. In A. W. Pink's words, God arranged the fall.
"no Adam did not have a free will after the fall."
Care to show where God said Adam did not have a free will after the fall?
Gen 3:17 Then to Adam He said, "Because you have heeded the voice of your wife, and have eaten from the tree of which I commanded you, saying, 'You shall not eat of it': "Cursed is the ground for your sake; In toil you shall eat of it All the days of your life.
Gen 3:18 Both thorns and thistles it shall bring forth for you, And you shall eat the herb of the field.
Gen 3:19 In the sweat of your face you shall eat bread Till you return to the ground, For out of it you were taken; For dust you are, And to dust you shall return."
"But, even before the fall, Adam operated within the purview allowed by God, so it was not totally free."
So if Adam did not have a free will prior to the fall then God made him sin, is that what you are saying?
And further if Adam did not have a free will after the fall then how was it the Cain did. Or are you going to tell us God made Cain kill Abel?
Gen 4:6 So the LORD said to Cain, "
Why are you angry? And why has your countenance fallen?
Gen 4:7 "
If you do well, will you not be accepted? And if you do not do well, sin lies at the door. And its desire is for you, but you should rule over it."
Gen 4:8 Now Cain talked with Abel his brother; and it came to pass, when they were in the field, that
Cain rose up against Abel his brother and killed him.
Gen 4:9 Then the LORD said to Cain, "Where is Abel your brother?" He said, "I do not know. Am I my brother's keeper?"
Gen 4:10 And He said, "
What have you done? The voice of your brother's blood cries out to Me from the ground.
Gen 4:11 "So now you are cursed from the earth, which has opened its mouth to receive your brother's blood from your hand.
Gen 4:12 "
When you till the ground, it shall no longer yield its strength to you. A fugitive and a vagabond you shall be on the earth."