The best solution would seem to use your wife's citizenship to gain the right to reside in Germany. You could conceivably try to apply for a work visa, “highly skilled migrant” visa or something like that but that's always a lot of trouble and I am not even sure if that's possible without a local employer sponsoring you.
I don't have any first-hand experience with this in Germany, but since your wife is an EU citizen I think you can just enter the country and apply for a residence card (Aufenthaltskarte). Following recent court decisions and changes in the law, this is not regarded as a “permit” anymore but is only declarative in nature. As the husband of someone making use of her right to free movement, you can reside in the same country and work without restriction. The card only documents this pre-existing right. If you do need a visa, it should be issued quickly and free of charge. All this assumes your wife is not German and does indeed fulfill the conditions (i.e. she must have a job, be self-employed or be able to prove she has sufficient means to support herself). Source: europa.eu.
If your wife is in fact German, then it's a completely different story. EU law does not generally apply and most people need to get a specific visa before entering the country and prove some knowledge of German (i.e. you can't enter on a regular Schengen visa and sort things out later). I assume that this is the procedure with a 3-month lead time you heard about. But since you are a US citizen, it doesn't apply to you, you don't need a visa or German-language proficiency test and can apply for the permit (Aufenthalstitel) from within the country:
Staatsangehörige Australiens, Brasiliens, El Salvadors, Israels, Japans, Kanadas, Neuseelands, Südkoreas und der Vereinigten Staaten von Amerika können die erforderliche Aufenthaltsgenehmigung auch nach der Einreise einholen.
Source: Ministry of Foreign Affairs
In any case, it will take several months before you get your residence card/permit but you can stay legally in the country in the meantime (you will get some document acknowledging your application). Since you don't need to apply in advance, the fact that you just married does not seem particularly relevant.
You should probably contact the nearest German consulate to verify all this. You could also contact the Ausländerbehörde from the place you are going to in Germany as you will need to deal with them after you move. They should be able to tell you what they can or cannot do and which documents they require in practice.