Super basic question about rhoticity

[Woods]

Do major dialects have all their syllable-final r's either unpronounced or pronounced, or are there places where r's are non-rhotic at the end of the word, but fully rhotic at the end of a non-final syllable?

That's how I speak, and I guess that's why to Americans I sound European and to Britons - American. So I was wondering if I have mixed in my pronunciation elements of very different types of English that never come together, or something similar exists somewhere.

I was listening to one British guy speak one of these days and liked very much the pronunciation, tried doing something similar for a couple of minutes (i.e. not pronouncing even word-internal syllable-final r's) but realised that feels extremely unnatural to me. At the same time, pronouncing r's at the end of the word feels strikingly North-American, very much exaggerated and definitely not something I want to do. 🤔