This is entirely within the rules. As an American citizen, you don't even need an EEA Family Permit. You could enter the UK without a visa and apply directly for a residence permit. There should be no problem.
In any event, if your EEA family member is with you, or in the UK, you have a legal right to enter the country under EU freedom of movement, regardless of the validity of your EEA Family Permit.
Also, I don't think you'll need to leave before the permit becomes active. If the Denmark trip is just a visa run because you're worried about being in the records as a tourist, just apply for admission at the border as the spouse of an EEA national and cancel the trip.
(When you cross any Schengen border with your husband, or if you are traveling to join him in the Schengen area, you are entitled to use the EU passport queue; this is explicit in the Schengen Borders Code. In the UK, I think it's less clear, but it seems that at least if you are traveling together you can use the EU queue.)
In response to ouflak's skepticism, here is some guidance from September 29th of last year: https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/464872/Direct_Family_Members_v3_0.pdf
It states in part that
An EEA or Swiss national and their direct family members’ right of residence does not depend on holding a document issued under these regulations.
It also says
[I]t is possible for [non-EEA family members] to demonstrate their right of admission under EU law at the UK border if they do not have [an EEA family permit, residence card, or permanent residence card].