A 1.1 Abitur is phenomenal! Congratulations!!!
Imo the main obstacle here is that you’re about to be done with school. As you probably know, you would have to take a gap year to apply to American universities for entry in 2019. I think there’s some chance that you could get sufficient funding to attend an American university, but I do not know if the chance of success is high enough to put your education in Germany on hold for a year.
Your options for funding from American universities break down into need-based aid and merit-based scholarships.
Need-based aid is available from the private “elite” universities (Harvard, MIT, etc) as well as liberal arts colleges. In your shoes, I would probably not apply to the elite universities. Your chances without something “extra” in addition to grades and test scores are too low to make that a worthwhile effort. (That something “extra” could be an award from an international academic competition, athletic ability, civic engagement or a particular hardship that you’ve had to overcome.)
A few dozen liberal arts colleges give significant financial aid to international students and they are generally less selective than the elite universities. Applying to these is still a gamble, but one that’s more likely to pay off than applying to Harvard & Co. Particularly if you are female and willing to attend a women’s college.
However, liberal arts colleges also come with downsides. You’d be very restricted in your choice of major, your degree would have no international name recognition, and you’d probably struggle to get sponsorship for a work visa in the US after graduation. (I am assuming that you’re not a US citizen. If you are, the whole funding situation changes, so please let us know!) If you attend a liberal arts college, you should be planning to get a graduate degree after your Bachelor’s.
The other option would be merit-based scholarships. There are a few scholarships out there for a full ride (tuition, room and board) that are based solely on grades and test scores, but they are at universities that you would not want to attend.
Full-tuition scholarships are more doable on the basis of grades and test scores alone. Here’s one such offer, as an example: https://www.uah.edu/admissions/undergraduate/financial-aid/scholarships/merit-tuition-scholarships You’ll notice that the University of Alabama at Huntsville makes a distinction between a 4.0 GPA (meaning perfect grades) and grades below a 4.0 GPA. If you haven’t had a single grade below a 2 (on the 1-6 scale), you can get your German transcripts evaluated by WES to a 4.0 GPA to qualify for the more generous scholarship.
If you take an offer for a full-tuition or a full-tuition + housing scholarship, and you’d need additional support for living expenses, you could get a student loan. I hear that Bildungsfonds are willing to give loans for a full degree abroad.