Thank you for sharing your heart and for asking for prayer. Your request for healing from cancer and every disease is received with love and taken seriously before the Lord, and you will be remembered in prayer.
Scripture presents Jesus as one who cares about the whole person. Matthew, writing under the guidance of the Holy Spirit, points to the prophecy of Isaiah and shows that what Christ bore includes physical healing as well as the healing from sin. When Jesus took bread and said this is my body which is broken for you, do this in remembrance of me, He was pointing to that offering of Himself for us. The Gospels tell us that Jesus went through the cities and villages teaching and preaching the good news of the kingdom and healing every kind of sickness and disease among the people. Large portions of the Gospels are given to those acts of mercy, touching and restoring those who were sick and afflicted.
In the law God even made provision for the cleansing of a leper, a disease that was loathsome and humanly incurable and that in Scripture often pictures the way sin spreads and deadens. Leviticus gives a law for the leper in the day of his cleansing, which shows that God left room for His own sovereign work to heal what no human could cure. When Jesus cleansed a man with leprosy He told him to go and show himself to the priest as a testimony. God still makes a way to work in cases that seem impossible to us.
The Lord reveals Himself as the One who heals. He becomes to us what we need, and for physical need He is the healer. I believe God heals today and that His power to heal has not ended. At the same time the Bible is honest that we do not understand all the ways of healing. There are gradual healings and there are processes of healing that God has built into our bodies, and all healing in that sense is from Him. The gifts of healing are given as the Spirit wills, and not everyone has the same gift, just as the body has many members with different functions. We can trust His ability and His compassion while also trusting His wisdom when answers come differently than we hope.
You mentioned the words in Third John about prospering and being in health even as the soul prospers. That was the heartfelt wish of John for his friend Gaius, not a blanket promise that guarantees immediate health for everyone, but it does show a real connection between the health of the soul and the body. Proverbs says a joyful heart does good like medicine, and we now know how much the inner life affects physical well being. To care for your soul, to keep your mind and heart nourished in Christ, is part of caring for your whole person.
Please be assured of prayer for you, that the Lord would touch your body, bring healing according to His perfect will, grant you peace and strength each day, and be near to you in every step. May the Lord bless you and keep you, make His face shine upon you and be gracious to you, lift up His countenance upon you and give you peace, as in Numbers 6:24. May you know the shelter of the Most High in Psalm 91 and the authority and joy that Jesus gives to those who belong to Him in Luke 10. Our prayer is that you would know His presence, His comfort, and His healing hand, and that whatever He does will be a testimony of His grace in your life.