Learn Russian: Я должен работать.

Not quite! While it does a similar job expressing obligation (must / have to), должен is actually a short adjective in Russian. This means it doesn't change its endings the way a normal verb does. Instead, it changes based on the gender and number of whoever has to do the action.

If a woman is speaking, would this sentence change?

Yes! Because должен behaves like an adjective describing the speaker (Я), a female speaker must use the feminine form and say Я должна работать. If a group of people were speaking, they would use the plural form: Мы должны работать.

Why is работать in its dictionary form instead of matching the speaker?

Just like with the modal verbs хотеть and мочь from this lesson, the word должен is followed directly by the infinitive (the dictionary form) of the action verb. You only change the first word to match the speaker, so работать (to work) stays exactly as it is.

Is there a missing word for 'have' or 'am' here?

In English we say 'I have to work', but the literal Russian translation is closer to 'I [am] obligated to work.' Since Russian always drops the 'to be' verb in the present tense, there is no word for 'am' here. You just put Я directly next to должен.