Hi, I'd like to help you with your question.
Although you have taken the very healthy step of quitting smoking, you probably have uncovered a psychological issue you have had most of your life. When you smoked, you demonstrated a self destructive tendency. You knew that your father and grandfather died from cancer but you still continued the habit. But recently, when you were able to quit smoking, you started on a path of good health but you still retained the self destructive tendency. But instead of being able to express this self damaging trait through smoking, now that you are a non-smoker, it now has no place to go. Because of this, you have developed an anxiety about your own health.
You have become convinced that you are going to be sick, even against what your doctor says and evidence you have seen to be true. But you continue to search for someone somewhere who will back up what you are sure is true. But no matter what you find, you are going to still feel these feelings because the basis for them has not been dealt with.
The first step in helping yourself overcome the anxiety and fear is to talk to a therapist. A therapist can help you pinpoint what in your life is causing these feelings, where they originated, and how to process them so you can cope better. To find a therapist, talk to your doctor about a referral. Or search on line at http://therapists.psychologytoday.com/rms/.
The next step is to challenge how you feel. Start writing down your beliefs about getting sick. For example, exercise will not make a difference. I will get sick anyway. Then find all the evidence you can to refute this. The idea is to change your beliefs by changing your thought process.
Next, you can practice thought stopping. This is similar to changing your beliefs but more immediate. Each time you are convinced you are going to get sick, say "stop" or use another technique such as a rubber band on your wrist. This not only makes you aware of your thoughts and how often you have them, but it provides a negative response to your thoughts.
And finally, learn to understand anxiety better. It will help you respond to your fearful thoughts and calm yourself when you feel upset. Here are some resources to help you:
The Anxiety and Phobia Workbook by Edmund Bourne
I hope this has helped you,