Hah yes! I grew up on the East Coast of the US and when I first moved here it was a physical effort to slow down my speech. If I spoke to way I used to, I was inevitably faced with a dismissive, 'I cah-n't understand a wuh-rd you're saying!'
I've been in London since 2006 and the way I made friends and fit in (to the point of meeting and marrying my English husband) is that I don't let being American define me. Yes, inevitably, people are going to comment on and bring it up, but if you act like it's just one of many aspects of who you are, just like where you went to school or where you work is just one aspect, most people get passed it and start thinking of you as you, rather than a representative of the US.
It's not that you have to hide where you're from, it's just that you need to act like it is what it is: one of many interesting things about you. Honestly, weeks go by in my office when my nationality doesn't come up as a topic.
Another tip that really helped me is that, when it comes to people being rude, unless explicitly stated, you can't assume it's because you're American. At first, I'll admit, was almost looking for offence on this front and it drove me crtazy. But I came to learn that you can't survive in a foreign country with that mentality. I think we can all agree that customer service here is beyond terrible. On top of this, people who work in in the service industry have bad days. The fact that a waitress/shop assistant is rude to you very likely has more to do with the fact that they just got dumped by their boyfriend than your country of origin. Thinking anything else leads to feeling really paranoid and, ultimately, isolated.