If your husband is a “worker” (more on that below), he has a right to reside in the country and you have a right to join him. This also means that you have the right to work in Germany as well, as a consequence of his right to live with his family. So being able to prevail yourself of this “family member” status does indeed depend to some extent on his finding a job.
Just about any job is enough to qualify as a “worker” under EU law. Technically, merely looking for a job would too but that cannot last forever and if he wants to do that longer than 6 months, he could have to show he is still actively looking for work and has decent chances to find some. In practice, if he was living alone and would abstain from applying for help from the state, he would not necessarily need to worry about this (see Moving to Germany with fiancée (both EU citizens) because of work reasons. What if she does not find a job?). But he will need to produce some evidence of his status when applying for your residence card.
Where things get a little complex is that, formally, you have the right to stay and work in Germany by virtue of living with your EU citizen husband making use of his treaty rights to free movement (a mouthful, I know). Under EU and German law, the residence card is supposed to document your pre-existing right, it's not a permit you would need to apply for.
So, theoretically, you could start to work immediately and worry about formalities later. It's up to the authorities to demand proof that your husband fulfils the conditions and, provided your husband does in fact fulfil them, even failing to apply for the required residence card after the first three months should only result in “proportional” sanctions (i.e. a fine, not deportation/ban).
But in practice, doing it this way seems a bit risky. I imagine the authorities will be reluctant to issue a residence permit in these conditions and if your husband does not find work within 6 months, you would find yourself in a very unpleasant situation.
Alternatively, if you have enough savings, your husband could also simply qualify as an “economically non-active person”. One advantage is that you could be able to do that immediately, without waiting for him to find a job. This requires showing that he has “sufficient resources” (i.e. income and/or wealth that would put you above the threshold to receive welfare benefits). I don't know the exact amount for Germany but it should be on the order of €10000-15000 for a year and he will also need to show that you both have health insurance. See also Can We Move To France Although We Don't Live In The EU?
If your husband does not fulfil any of these conditions, 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 legally speaking. 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 cannot work without authorisation.
Finally, note that you will actually need two distinct “registrations” in this process. I described the requirements for a residence card or permit but in Germany, unlike the US or Ireland, everybody (including German citizens) have to officially register their address within one or two weeks of moving somewhere (whether it's one week or two depends on the state).
This address registration is completely unrelated to the 3-month threshold and you need proof of it (Meldebestätigung) for many official purposes, including applying for a residence card so both of you should do this as soon as possible in any case. You get the residence card from the Ausländerbehörde, whereas the Meldebehörde is in charge of the regular address registration.