John 6:44 reads No man can come to me, except the Father which hath sent me draw him: and I will raise him up at the last day.
But notice that verses 45 and 47 tell HOW God draws a sinner:
It is written in the prophets, And they shall be all taught of God. Every man therefore that hath heard, and hath learned of the Father, cometh unto me … Verily, verily, I say unto you, He that believeth on me hath everlasting life.
So this passage does not teach that a sinner cannot act. To the contrary, it actually teaches just the opposite, that a sinner must be taught (hear) the gospel, learn it, and believe it!
Yes, God does draw sinners, through His word! → Romans 10:17:
So then faith cometh by hearing, and hearing by the word of God.
We are called by the gospel. – II Thessalonians 2:14