In Azuma's Inequality, is the statement true when $X_k  X_{k1} < c_k$ almost surely rather than with probability 1? If not, is there another result which gives strong concentration when the above inequality (for each $k$) holds with high probability?

There is a large literature on variations of Azuma's inequality. One lemma that is similar to what you ask is Lemma 3.1 of this old paper of Wormald and myself. It considers the case where $X_kX_{k1}$ is within one bound with very high probability and within some wider bound always. There are lots of such results. 

