Thank you so much. What is happening when I try to practice mindfulness is I am reaching concentration on breathing spontaneously instead of observing the rising and falling. The accompanying cessation of breath is the result of long term practice of Pranayama and other kinds of yogic meditation. The state is stillness, which is a desired yogic state.
But the issue is there are some mental defilements (kleshas) and observing them will give me some ability to modify my thought patterns.
Once I get out of stillness, I notice I am still (but no longer still) , so it is indeed "thinking" after that point. This is my third day of practicing mindfulness. I will give it some more time. I feel like I am learning a new skill like cycling. I am bound to fall many times as I try to find the balance.
Thanks for your detailed answer. Will read the link too for any doubts.