The Directed Acyclic Transformer is a fast non-autoregressive (NAR) model that performs well in Neural Machine Translation. Two issues prevent its application to general Natural Language Generation (NLG) tasks: frequent Out-Of-Vocabulary (OOV) errors and the inability to faithfully generate entity names. We introduce Control-DAG, a constrained decoding algorithm for our Directed Acyclic T5 (DA-T5) model which offers lexical, vocabulary and length control. We show that Control-DAG significantly enhances DA-T5 on the Schema Guided Dialogue and the DART datasets, establishing strong NAR results for Task-Oriented Dialogue and Data-to-Text NLG.