Having the freewill to make a choice doesn't mean God didn't do the work to enlighten us to make that choice. The Bible teaches the Holy Spirit convicts us of our sin and unrighteousness and need for the Saviour - but man is still given the freewill to choose to accept or reject the Lord Jesus Christ and His salvation - which I have shown in various passages.

If drawing in itself meant they would be saved (according to your theology), then everyone would be saved - because Jesus said if He was crucified He would draw all men unto Himself. I believe it means that He would give everyone an opportunity to hear the Gospel and have the choice to be saved.