For me personally, I have found the opposite happening. I feel through the years God has given me the courage and strength to bear so much more suffering and experience so much more joy. He has expanded my emotional perimeters in all directions. I cry for joy and I cry for sorrow much more easily. My emotions are free from embarrassment--no longer shackled by callousness. God's grace has given me a capacity to bear so much feeling that before I would have run from. Now I am able to truly cry with someone needing me to cry with them and at the same time feel enormous strength not to cave in to the strong pain.
Rather than becoming less rational, I feel I can now truly see reality, the pain and chaos of reality and rather than run from it and stupefy and benumb myself with entertainment, I can face it head on and take the punch. I hurt more, but I live more. And even in pain I have this unearthly capacity to understand joy at the same time. Suffering no longer terrorizes me, it no longer controls me. Now I am freed to feel anger and joy and sorrow, but I feel much less fear.
Emotions are not the enemy, they are a heavenly gift. Becoming more like Christ doesn't mean you will feel less, but feel more and His grace will sustain you through it all. Emotions often are used by God to drive us to Him and to drive us to doing the right thing. So do not try and turn them off. Offer up your pains and sorrows as a great sacrifice to Christ. You will be able to move about in the darkest parts of humanity and feel their pain and yet continue help them with courage.
Let Him expand your capacity to feel life. It will give you more and more capacity to give to others who do need not only your physical help, not only your logic but your heart.