If your husband does not fulfil any of these conditions (being a worker, a student or having sufficient resources), then you need to qualify for the relevant German work permit on your own. Being married to an EU citizen would not make much difference. That's usually harder but for an academic job, it should be doable. In this case, you definitely need the permit/authorisation before starting to work.
US citizens who are not married to EU citizens can also enter the country and stay for three months without a visa but. They can also apply for a residence permit and stay as long as their application is pending. But, unlike people covered by the EU freedom of movement, they are not allowed to work without prior authorisationbefore having received the permit that explicitly allows them to do so.