In this paper, we investigate problems of communication over physically-degraded, state-dependent broadcast channels (BC) with cooperating decoders. Three different setups are considered and their capacity regions are characterized. First, we study a setting where noncausal state information is available to the encoder and the strong decoder. Furthermore, the strong decoder can use a finite capacity link to send the weak decoder information regarding the messages or the channel state. Second, we examine a setting where the encoder and the strong decoder both have access to noncausal state information, while the weak decoder has access to rate-limited state information. This scenario can be interpreted as a special case, where the strong decoder can only cooperate to send the weak decoder rate-limited information about the state sequence. A third case we consider, is a cooperative setting where state information is available only to the encoder in a causal manner. Finally, we discuss the optimality of using rate-splitting when coding for cooperative BC. In particular, we prove that rate-splitting is not necessarily optimal when coding for cooperative BC and solve an example where a multiple-binning coding scheme outperforms rate-splitting.