This sounds like a version of what’s known as the pin/pen merger, or some closely related vowel merger. It’s one of various phonemic mergers in English — processes where in certain accents, in certain contexts, phonemes which were previously/elsewhere differentiated merge to be pronounced the same.
The aforelinked Wiki article has some nice stuff — in particular, it has a nice map of the rough distribution of the pin/pen merger in the US:
For more detail, there’s also an excellent discussion on Language Log. On the whole it defies summary, but two important take-home points are:
these things are notoriously difficult to accurately understand by introspection. What we say often isn’t what we think we say — it really feels to me like I pronounce wine and whine differently, but I sat down with a friend and a tape recorder once and reluctantly admitted that apparently I say them indistinguishably (just as linguists tell me that almost everyone of my geographical/cultural background does).
for this and more reasons, written discussion of these things can easily get extremely unenlightening — we can ask each other “so do you also pronounce pig like peg?” but we and anyone reading will all have different ideas of what each of these pronunciations would mean, and none of those may correspond to what any of us actually say! See the comments on the LL post above for many, many examples of this phenomenon…
(I am not a trained linguist, by the way — so anyone more knowledgeable, please feel free to correct any of the above if I’ve gotten something wrong.)