Well, speaking as one of the very few born and bred Yanks here on this board, I can 100% reassure that I have never offended any British as a tourist, since I've been never been fortunate enough to travel overseas, much as I'd like to.
(Now offending people here is a totally different thing!
My total anglophilia alas doesn't from direct experience of the UK but from my experiences having befriended and worked with many British people from childhood on. I can't think of a one of them who wasn't charming, funny, and cultured.
As for British TV, yes, like many Yanks I grew up on all of the classic Brit comedies and dramas of the 70s and 80s. I tend to prefer the more intelligently written shows (Python Fawlty Towers, Yes, Minister, Red Dward) over the more mundane nudge-nudge wink wink stuff (Are You Being Served and all that).
But I'm also an avid reader of British and Irish authors, from the classicists like Swift, Fielding, Hume and Johnson to the 20th century comic writing greats like Woodhouse, Waugh, both Amises, and of course that great half-American, Winston Churchill.
Personally, if I ever went to England I would be the antithesis of the obnoxious Yank tourist because I'd be too much in awe of just being there.
In terms of who gets what shows, it amazes me that you Brits get so many of our (mostly terrible) TV shows almost as quickly as they're broadcast. In the "olden days" it tooks several years for our shows to cross the ocean, and you'd only get the good ones. When I hear Brits talking about Two and a Half Men and How I Met Your Mother, well, frankly, I'm astounded. Here, the shows are considered to be middlebrow puffery. Highly rates middlebrow puffery.
Yet, the same flow of shows from Britain doesn't come here. Other than BBC America (which broadcasts mostly reality shows and very few new comedies) you might occasionally see a British comedy airing on a public station, but mostly it's the middle of the road shows like AYBS and shows about retired people. I really wish there was a dedicated cable channel here that would broadcast the newest and edgiest British comedies. I feel we're missing out on the real good stuff.