There's a term I've heard, "prevenient grace", that was defined something like a grace given to us that enables us to choos whether we will believe or not. I think this agrees well with Titus,T - Total Depravity: Calvinism teaches that man is totally depraved and cannot make a decision to receive Christ unless he has already been born again of the Holy Spirit.
I would teach that there is a middle ground between total depravity and being born again...being drawn by the Holy Spirit...in which a man, who is not yet regenerated or born again, is enabled to either receive or reject Christ in that window of opportunity in which the Holy Spirit is drawing him. If he then receives Christ rather than rejecting Him, he becomes born again of the Holy Spirit through faith in Jesus Christ.
Verses that substantiate this are Ephesians 2:8-9 and Romans 5:1-2.
Titus 2:11-12
11) For the grace of God that bringeth salvation hath appeared to all men,
12) Teaching us that, denying ungodliness and worldly lusts, we should live soberly, righteously, and godly, in this present world;
JP Green translates this as "For the saving grace of God has appeared to all men." I think this refers to a measure of faith being given to each, and that we have to choose whether we will "mix our faith with the Word",
Hebrews 4:2
For unto us was the gospel preached, as well as unto them: but the word preached did not profit them, not being mixed with faith in them that heard it.
I think that this idea of total depravity is right, that mankind is fatally corrupted, and unable to seek after God. But that God gives us a grace to enable us to choose to mix the faith He gives to us with the Gospel message we receive.
Much love!