This sounds as if you're trying to say the word total doesn't mean totally. Being totally depraved would mean un-saveable. Wouldn't it?Total Depravity doesn't teach that unsaved people cannot do "good" things. Total means all of your has been affected by sin, not that you are as evil as you can be. I'm sure Hitler did some "good" things.
Before Salvation we are lost with no hope of Salvation until that hope is given us. Hope is given us by our being convinced of the gospel. Once convinced by God's messenger and the Holy Spirit we believe because we are convinced and that belief is faith. Yet faith can only save if we submit to the righteousness of God. Rom 10:1-4
These Pharasee's in Romans believed in God but they could not submit to God's righteousness of sending His Son to die for the sins of the whole world. They rejected Christ.
Submission is not a work it's giving up to Christ.
MB