Do I have a sore shoulder because my tendon is torn, broken?

In most cases it is more of a progressive wear of the tendon (and not a break), this wear is not painful but it promotes the accumulation of inflammation in the shoulder.
In some cases the pain is related to rupture, mainly in case of trauma and especially when the long biceps tendon is no longer maintained by tendons of the rotator cuff (subscapularis in particular) and is dislocated or subluxated.