The very fear you describe, that women are always judging you and will laugh at you, is itself a form of unbelief. You are more afraid of their opinion than of offending God. Consider the Israelites who stood at the borders of the promised land: they had seen God's mighty works, yet they yielded to fear and imagined nothing great of Him. By that unbelief they perished in the wilderness. Do not follow that same example. You say you do not want to feel this way, and I believe you. But your avoidance of women in public is not humility; it is a proud fear. You assume you know their thoughts, and you let that dictate your steps. Lay that pride low. Be not high-minded, but fear, fear God, not man.
The yoke of Christ is easy, and His burden is light. Learn from Him who is meek and lowly in heart, and you will find rest for your soul. When you see a woman, remember that she is a fellow creature made in God's image, and that Christ died for her as much as for you. If you harbor resentment or suspicion, you are drinking from a cup that is not the Lord's. You cannot partake of the Lord's table and the table of demons, the table of envy, suspicion, and hatred. That path leads to spiritual ruin. Instead, imitate the faith of the centurion and those who brought the sick to Christ. Even when you cannot approach another person without fear, the faith you exercise in prayer can bring healing to this deep wound. Cry out to the Lord as the woman with the issue of blood touched the hem of His garment: her faith made her whole, and so yours can drive out this unclean spirit of misogyny.
You mention family members who do not believe, and one who treats the Scriptures lightly, even claiming what God calls sin is not sin. This should grieve you, but do not let their unbelief infect your own. God often allows temporal troubles, fears, losses, conflicts, to wake us up before the final judgment. Do not be like Sarah in her laughter of doubt, but be like her when she feared and said, "I did not laugh." Let holy fear drive out the remnants of unbelief. Pray earnestly for your family, and guard your own soul. The Lord delays judgment out of lovingkindness, so that sinners may wash away their transgressions. But that window will not stay open forever. As for you, do not waste your days hiding from shadows. A garment your servant wore you would not touch, yet you clothe your mind with the filthy rags of suspicion and avoidance. Be ashamed of the sin, not of the words I speak. Let the fear of God cast out the fear of man, and walk in the freedom of those who know that Christ has burst the gates of hell and loosed the dead. He can certainly loose you from this bondage. Cry out to Him, and do not delay.