What is the ruling on my leaving a mosque due to the Imam prolonging his speech, and my going to another mosque to catch the congregational prayer, and do I bear any sin for that, and what is your advice to me?

It was more proper for you to wait in the mosque to pray with the Imam, because departing from the mosque after the Adhan is disliked (makruh) or forbidden (haram). Had the Imam prolonged the prayer, he could have been gently advised after the prayer. Some scholars have permitted leaving to pray in another mosque if you know that you will catch the congregational prayer there, but this should not be done. We advise you not to repeat this action and not to leave the mosque after the Adhan except for an excuse.