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 avail yourself of this “family member” status does indeed depend to some extent on his finding a job.
In practice, you would receive a “certificate of application for the residence card” as soon as you apply for a residence card, which should in any case be enough to establish your status for the first six months after the application so you don't need to wait for the paperwork to be processed to start working. But you do need to submit proof of your husband status (e.g. a work contract) when applying for this residence card.
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 of finding some. In practice, if he was living alone and would abstain from seeking support 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 without a job, it will be harder to prove 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. As long as your husband is working or looking for work, he is making use of his treaty rights and under EU and German law, the card you would receive in this case is supposed to document your pre-existing right to stay with him. It's not a permit you would need to apply for in advance. That's why it's called an Aufenthaltskarte (or, in the UK, a “residence card”) and not an Aufenthaltserlaubnis (literally “residence permit”).
Theoretically, you could therefore even start working immediately and worry about formalities later. It's up to the local authorities to demand proof that your husband fulfils the conditions and I have no idea whether your employer could or would check your status. Provided your husband does in fact qualify as a worker, even failing to apply for the required residence card during the first three months should only result in “proportional” sanctions (i.e. a fine, not deportation/ban).
But doing it this way seems a bit risky, especially if you cannot prove that your husband already has a job. I imagine the German authorities would be reluctant to issue a five-year residence card in this situation and if he does not find work within a reasonable time, you would find yourself in a very unpleasant situation. So if is the goal is to keep things simple and avoid bureaucracy, I would pursue other options first.
For example, if you have enough savings, your husband could also simply qualify as an “economically non-active person”. One advantage is that he could be able to prove that immediately, without waiting to find a job. You would then qualify for the same “family member” status and would thus be allowed to work in Germany without restrictions.
Applying for a residence card as the spouse of an economically non-active person requires showing that the sponsor has “sufficient resources” (i.e. income and/or wealth that would put you above the threshold to receive social benefits). I don't know the exact threshold for Germany but it should be on the order of €10000-15000 for a year and you will also need to show that you have health insurance. See Can We Move To France Although We Don't Live In The EU? for more details on how these rules work in another EU country.
Importantly, being economically non-active means that there are additional requirements regarding income and insurance to be met to sponsor your residence card but your husband could still take up employment or start a business at any time.
For completeness, there is a third category of people (namely students) who have a right to reside in other EU countries under free movement rules but that's presumably not relevant for you and your family.
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. 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 before having received the permit that explicitly allows them to do so.
Finally, note that you will actually need two distinct “registrations”. I described the requirements for a residence card/permit but in Germany, unlike the US or Ireland, everybody (including German citizens, who obviously don't need any residence permit) is required to officially register their address within one or two weeks of moving somewhere (whether it's one week or two depends on the state). Failure to do it in time can result in a not-insignificant fine.
This address registration requirement is completely unrelated to the 3-month threshold or to your citizenship and status as a foreign citizen. You need proof of it (Meldebestätigung) for many official purposes, including applying for a residence card so both of you should do it 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 (depending on the town, the Meldebehörde can be called Einwohnermeldeamt, Bürgerbüro, Bürgeramt or something else along those lines).