Man's free will does not depreciate God's sovereignty, man calls on the name of God then God performs His divine work on them and receives all the glory. The Holy Spirit reveals the truth of His word to man and convicts of sin and man becomes enlightened by His word, man confesses their sins and surrenders to God, and God purifies and transforms of their heart, justifies, saves, and sanctifies; man cannot be justified without Him. God is no respecter of persons that's why out of His sovereignty has given man ability to choose.
My understanding of total depravity is that man is born into this world enslaved by sin however man is not completely dysfunctional that he cannot do anything good or call on God to be saved, For whosoever shall call upon the name of the Lord shall be saved (Rom 10:13). God created man in His own image, innocent and sinless, but the sin of one man marred humanity thus the unregenerate man, who has not tasted the gift of Christ's righteousness, is not righteous before God (Rom 3:10). The righteousness of a sinner is as filthy rags before God (Isa 64:6). This is why Jesus paid the ultimate price for our righteousness and the sinner needs to be saved and have God's righteousness bestowed upon them by faith so that genuine righteousness that comes as a fruit of regeneration will be pleasing to God, Even the righteousness of God which is by faith of Jesus Christ unto all and upon all them that believe: for there is no difference (Rom 3:22).
God permits man to accept or reject His plan of salvation and those who reject it will face the second death. Jesus instructed that we go into the world preaching the gospel- the good news of His death, burial, and resurrection for the salvation of mankind- so Christ provided a way of salvation for everyone, not just for an elite group, The same came for a witness, to bear witness of the Light [Jesus], that all men through him might believe (John 1:7); And the Spirit and the bride say, Come. And let him that heareth say, Come. And let him that is athirst come. And whosoever will, let him take the water of life freely (Rev 22:17).
Limited atonement and unconditional election is unbiblical. God is not willing that any should perish but that all should come to a knowledge of the truth. John 6:51- I am the living bread which came down from heaven: if any man eat of this bread, he shall live for ever: and the bread that I will give is my flesh, which I will give for the life of the world. John 1:29- The next day John seeth Jesus coming unto him, and saith, Behold the Lamb of God, which taketh away the sin of the world. The just God loves all mankind that's why He sent His Son to offer Himself as a sacrifice for all. His will is for all to be saved but His will does not always come to pass (this would be His permissive will); not everyone will be saved because not all exercise faith in Jesus and many are choosing darkness rather than light. If God arbitrarily predestined some to be saved and others to be lost without giving them an opportunity to be saved and if Jesus died for only the elects, then it would be injustice to hold the non-elects accountable for their sins. However, God is just and if anyone lands in Hell fire it is their fault, not God's because He provided a way of escape for all but not everyone choose that way.
For many are called, but few are chosen (Matt 22:14)- Those who respond to the unlimited invitation by repenting and believing in Him will be chosen.
How shall we escape, if we neglect so great salvation; which at the first began to be spoken by the Lord, and was confirmed unto us by them that heard him (Heb 2:3)- One won't escape the second death if they neglect the gift of salvation. The fact that salvation can be neglected proves that man has free will.
Behold, I stand at the door, and knock: if any man hear my voice, and open the door, I will come in to him, and will sup with him, and he with me (Rev 3:20)- Jesus stands at the door of man's hearts and knocks, and when they open up their hearts, He comes into their life. The fact that man opens the door of their hearts proves free will.
Total depravity is not an excuse for gallivanting around with sin after salvation. One who has been genuinely set free from sin is no longer in bondage to it as they are dead to sin and alive in Christ, and they strive to walk in the Spirit by His power. However, the believer is still a free moral agent so they can either continue on the narrow way by loving the Lord and obeying His word or depart, And if it seem evil unto you to serve the LORD, choose you this day whom ye will serve; whether the gods which your fathers served that were on the other side of the flood, or the gods of the Amorites, in whose land ye dwell: but as for me and my house, we will serve the LORD (Josh 24:15). The reality is that not all saints persevere to the end and many backslide during the course of their walk with God. Demas and Hymenaeus left the faith and the Holy Spirit said that in the latter days many will depart from the faith heeding to seducing spirits. It is not God’s will that we fall away but He wants us to, by our own free will, serve Him in sincerity and truth, feed ourselves with His word in order to grow, and make our calling and election sure. There is security in Jesus Christ and we have God's enabling grace and guidance from His Spirit to keep us on the right path.