I've set my comic in a fantasy universe, so I don't have any real-world regional accents in it - differentiating between characters, I focus more on individual mannerisms and levels of formality. I've got one character who speaks very brusquely, another who stays very formal, one who flares up and gets angry very easily, etc.,.
In general, I've found that less is more, when reading comics that do. If you write the accent out all phonetically, trying to mark out every quirky vowel and dragged out consonant, it becomes a pain to read. Put just enough phonetic spelling in for the reader to realise there IS an accent - picking certain dialect-words instead of standard ones is good too, as is using any regional grammar-rules (some turns of phrase are more common in certain dialects, etc.).
Just try to keep things as readable as possible. A few dropped g's and h's work fine, but when you start changing the spelling of every single word, it becomes quite difficult to understand what they are even saying, especially for those of us who speak English as a second language.
I think focusing more on word-choices than on phonetic spelling might be good. Consider which words are used in which region - the debate on whether to use "soda" or "cola" or "pop" to describe carbonated drinks, for example, or which variant of "you" people use (you/y'all/yourself, etc.) - and concentrate on word-order and grammar choices. Some phonetics are fine, as I said, but just don't go overboard.
This may be true for some people, but not for everyone. I speak English as a second language, and I speak it so fluently that I'm MORE likely to use contractions and combined words.
I learned it in a classroom, and am not a second generation immigrant. I'm a Swede, living in Sweden, surrounded by other Swedes. I might occasionally trip up on a word-choice or some grammar-rule, but it doesn't happen often. My fluency is a matter of practise and exposure to English-language media.
How a character speaks will mostly depend on who they are, rather than what they are, IMHO.