Learn Polish: Dzisiaj nie widziałem pani dziekan. - I did not see the dean f today.

High-ranking professional titles like dziekan (dean), profesor, or minister remain in their uninflected, masculine form when referring to a woman. Instead of declining the title itself to show its grammatical role, we decline the word pani that comes before it.

If we are supposed to decline pani, why does it look like it's in the Nominative case here?

It actually is declined! The verb widzieć (to see) normally takes the Accusative case. If the sentence were positive ("I saw the dean"), it would be Widziałem panią dziekan. However, because the verb is negated (nie widziałem), the object flips to the Genitive case. The Genitive form of pani just happens to be exactly the same as the Nominative: pani.

Can I drop pani and just say Dzisiaj nie widziałem dziekan if the context makes it clear she's a woman?

No, you shouldn't drop pani. Because the title dziekan doesn't change its endings to show grammatical case when referring to a woman, the sentence relies completely on pani to carry the case (Genitive) and to mark the gender. Without it, the sentence sounds grammatically broken.

If the dean were a man, how would this sentence change?