Speaking hastily
Though the jury is still out, I have been informed by a non-Californian (well, a deep-Southerner-turned-Californian) that I, in fact, do have several of the Great California Vowel Shift. Namely, the lowering and/or backing of front vowels before oral consonants. So I (apparently) don’t pronounce did like dead, but it’s (barely perceptibly) somewhere in between the two. I’ll have to surreptitiously record myself to find out, though I did notice that perhaps my low front vowel (the one in that) is usually laxer and further back than the “standard,” which is part of the perceived vowel shift.
And for something totally different, I present, for your viewing entertainment Stephen Pinker vs George Lakoff in a battle for American progressives’ hearts and minds.