A good model is Alice Walker's The Color Purple. I suggest perhaps you read it, or read it again if you have already. Other than a few brief passages that are written from the viewpoint of the MC's sister Nettie, most of this novel is the MC Celie's voice, and it is written exactly as she would speak, in the black dialect of a rural Southern girl. Somehow, Walker manages to pull this feat off without alienating the reader (and don't forget this was a bestseller and a Pulitzer Prize winning novel). But I think the real trick is in examing the craft of how she achieves this.
First of all, the novel is an epistolary narration, so we accepot from the beginning that we are reading not only the character's narration, but her written letters. This automatically gives some license.
Secondly, the effect is achieved by carfeully choosing to manipulate words in such a way that an average reader can still figure it out without having to put a lot of effort into it. An example: "She tried to get his
tension." We know immediately, without having to decipher, that what Celie is saying is that "She tried to get his attention."
Thirdly, there has to be consistency, so that once a reader has learned your character's "code" and has grown comfortable with it, they can safely assume it's not going to change (unless we switch to another character's voice). This means if the character says "git" for "get" on page 1, we can trust they'll do so on page 301 as well. (Of course, you can make some subtle allowances if it's understood that your character has gone through significant change by the end of the book--obviously, if they are an older person, the voice may have grown a bit more sophisticated and mature. Or if they've become more educated and well-read. Naturally their voice should reflect such changes, but most readers will aceept this if you give them reason to believe the change is credible).
Another great example is Huckleberry Finn. Both the narration AS WELL as the dialogue is written in Huck's voice, and it seemed to have worked for Mark Twain just fine.
I agree with those here who have said you have to be true to your character's voice, and this is as true of their narration and internal monologues as well as their dialogue. However, if you're finding that too many readers are being put off by it, you might try reeling the voice in just a bit and making some subtle alterations. For example, if you've resorted to using funny or phoentic spellings, that may be too gimmicky for most readers. You can get the idea of their voice across by their manner of speaking and the words that they use. Unique phrases are one way of achieving this effect: "She had two sons, but everyone could tell it was David--the youngest--that hung her moon." Readers are usually much more tolerant of that kind of thing than trying to decode an entire idiom.
But I would highly suggest reading as many works as you can find that are similar in tone and style to what you're doing, that is, first-person narratives where the MC is somewhat young, naive, or uneducated. Narrow those down to the ones you really like--the ones you find most readable--and assess how the writer achieves their effect. This is really the best way to learn, and as for the critque groups, take it with a grain of salt. Many people who offer that kind of critque simply haven't read enough, or they haven't read enough first-person narratives in this style that truly work. Remember that a good critique group will work WITH your story and the aims it is trying to achieve; not to try to turn it into something it's not.