Discourse adverbials are key features of discourse coherence, but their function is often ambiguous. In this work, we investigate how the discourse function of otherwise varies in different contexts. We revise the function set in Rohde et al. (2018b) to account for a new meaning we have encountered. In turn, we create the "otherwise" corpus, a dataset of naturally occurring passages annotated for discourse functions, and identify lexical signals that make a function available with a corpus study. We define continuation acceptability, a metric based on surprisal to probe language models for what they take the function of otherwise to be in a given context. Our experiments show that one can improve function inference by focusing solely on tokens up to and including the head verb of the continuation (i.e., otherwise clause) that have the most varied surprisal across function-disambiguating discourse markers. Lastly, we observe that some of these tokens confirm lexical signals we found in our earlier corpus study, which provides some promising evidence to motivate future pragmatic studies in language models