Learn Farsi: من بیمار هستم و باید دارو بخورم.

In Farsi, you do not use the verb for 'take' when talking about medicine or pills; you use the verb xordan, which means to eat or drink. Since the sentence is about what I must do, it takes the 'I' ending -am.

Why is the verb boxoram instead of the normal present tense mixoram?

The word bâyad (must) requires the verb that follows it to be in the subjunctive form. For this verb, that means dropping the normal mi- prefix and replacing it with bo-. So, mixoram (I take) becomes boxoram (that I take) after bâyad.

If the subject is 'I', why doesn't the word for 'must' (bâyad) have an 'I' ending?

The word bâyad is an impersonal verb, which means its form never changes. You use the exact same word for I must, you must, we must, and so on. You always know who must do the action because of the personal ending attached to the main verb at the end of the sentence (like the -am on boxoram).

Do I need to repeat the word man (I) after va (and) for the second half of the sentence?