Hey, Mike! Nice to meet you!
Do true believers have mental illnesses? Yes.
Do their mental illnesses slowly get destroyed by Christ? That depends. Secular psychology has deeply embedded itself in the Christian community. So much so these days that it is given greater weight in the minds of many believers than God's truth offered to us in the Bible. What is forgotten is that much of modern psychology is still a very soft "science" and obtains its credibility largely by association with harder, more well-established fields of science like neuro-chemistry, neurology and biochemistry. Modern psychology is still in its infancy and is boiling over with new theories about the workings of the human psyche, many of which are in contention with each other and most of which end up in the dustbin of failed theories. Modern psychology is also derived from sources that, by and large, are naturalistic in their philosophical disposition. That is, they rule God out entirely right from the start. It is no surprise, then, that the remedy the Bible gives for mental and emotional distress often differs widely from the answers arising from secular psychology.
The Bible is very clear that, barring some biological deficiency or injury-related cause of mental and emotional dysfunction, peace, and joy, and contentment, and mental stability can be found in fellowship with God (Isa. 26:3; 54:13; Jn. 14:26, 27; 16:33; Ro. 1:7; 8:6; 14:17; 1Cor. 1:3; Ga. 5:22; Phil. 4:6, 7; Col. 3:15; 2Ti. 1:7, etc.). Listening to the sage-sounding claims of modern psychology, many Christians have come to doubt this is true, however, and have resorted instead to endless drug "therapy" and "counselling." The drugs and counselling never actually provide a remedy but they do blunt the force of the problems people face within themselves and this is preferable, it seems, to getting honest with God, and accepting responsibility for one's thought life and emotional state, and walking rightly with Him. It is in doing these things that real inner peace is found and maintained.
Moving into fellowship with God and the peace and joy that is in Him takes time. It is definitely a process. As you continue to walk with God and go deep with Him, however, the "peace that passes all understanding will keep your heart and mind." Don't be turned aside from doing so by those who would tell you godless humans know better.
Isaiah 26:3-4
3 You will keep him in perfect peace, Whose mind is stayed on You, Because he trusts in You.
4 Trust in the Lord forever, For in Yah, the Lord, is everlasting strength.
Selah.