To enter the UK, you do not need any specific document, because you are from a "visa free" country. (If you were from a country whose citizens need visas for short visits, you would need a settlement scheme family permit.) You will, however, need to show evidence that you are married to an EEA national who is living and working in the UK.
To remain in the UK, you do not actually need any document as long as the UK continues to participate in the EU free movement regime. As of March 2020, this is expected to end at the end of this year. Under free movement, you have a legal right to be in the UK that supersedes any limitations that might be stamped into your passport, such as "leave to enter for six months" or "employment prohibited."
Although you do not need any document to remain in the UK, a document to demonstrate your right to do so can be useful in some contexts. Also, you will need a document to remain in the UK after free movement ends. That document is the one you'll get when you enroll in the EEA settlement scheme.
What do I need to move to the UK permanently and live (not work) there?
To move there, you'll need your US passport, your marriage certificate, proof of your husband's German nationality, and evidence that he is working in the UK. To live there after the UK stops participating in EU free movement, you'll need a settlement scheme document.