Final answer:
The Past Continuous Tense is used to emphasize the ongoing nature of an action. 'Have to write' is used to indicate an obligation or necessity.
Step-by-step explanation:
In the first sentence, the Past Continuous Tense is used to emphasize the ongoing nature of the action. This tense is used when an action was in progress at a specific point in the past. For example, 'I was writing an essay when the power went out.'
In the second sentence, 'have to write' is used instead of Future Simple tense to indicate an obligation or necessity. It means that writing the essay is a requirement or something that must be done. For example, 'I have to write an essay for my English class.'