'I must now cook' was rejected, but seems like a perfectly acceptable use of English.
It sounds a bit more formal than normal, but I'll add it. Here's an explanation you may find interesting:
Putting an adverb that tells us when at the end of a sentence is a neutral position, but these adverbs can be put in other positions to give a different emphasis. All adverbs that tell us when can be placed at the beginning of the sentence to emphasize the time element. Some can also be put before the main verb in formal writing, while others cannot occupy that position.
Because of the modal verb (muss). Only one verb modifies itself to match the person doing that action (only one verb conjugates to match the subject). When using a modal verb in present tense, the modal verb conjugates and the other verb goes to the end in the 'blank' form (called the infinitive).
ich koche = "I cook"
ich esse = "I eat"
ich muss kochen = "I need to cook"
ich muss essen = "I need to eat"
It's similar in English: We say "He needs to eat", not 'He needs to eats".