I am incredibly familiar with John Piper, Paul Washer, and to a lesser (but still knowledgable degree) of Mark Driscoll. And while I would not recommend Driscoll or Piper (unless, you listen to or read Piper's earlier works), I would highly recommend Paul Washer to anyone. The term you are referring to for Paul Washer is not neonomianism (that's a first for me!), but rather "Lordship Salvation" is what Mr. Washer teaches.
For Luke 9 reads, "23 And He was saying to them all, " If anyone wishes to come after Me, let him deny himself, and take up his cross daily, and follow Me. 24 "For whoever wishes to save his life shall lose it, but whoever loses his life for My sake, he is the one who will save it. [Luke 9:23-24]
This is the crux upon which the Gospel rests with us. Christ would say that His family is them who do the will of God (Matt 12:47-50). I agree that these works will not attain Heaven for anyone, because God freely elects whomever He will (Rom 9:16), but rather, it is critical of us to understand that in God's covenant to renew the hearts of His own, that He would give us hearts that not only could hear His voice (Jn 10:27), but hearts that would obey Him (Rom 6:1-7) [however mixed with failure; and it will be, for we are not free completely from sin!].
I hope I haven't further complicated things; but while I don't agree with ALL things Paul Washer says; I know that he is firmly Biblical and a man who, very well, understands grace.