It reminds me of that song–what’s the name? You know the song. It goes: da - da -da da – dum da dum da da dum - dada da da da. You know that song–right? Of course you do, because it’s obvious to me, so when I write it out like that with 'plain sounds," you clearly will recognize the same sounds that are in my head.
Yeah, same here as far as the sounds not really changing to schwas in those words in fast extemporaneous speech, so far as I can tell. (And, yes, a lot of times it is difficult to “hear” how you sound–you may think you’re making one sound consistently across contexts, but you may not be.)
That said, I’m pretty sure my pronunciations are thus: The vowel in “the” is usually /ə/. Sometimes, for emphasis, it’s /i/ (or /i:/, I’m not going to differentiate length here.) And very rarely, it might edge towards /e/ɛ/. “That” is almost always /æ/. It might edge towards an /e/, but never a schwa, so far as I can tell. “Than” in my dialect usually gets an /e/ and not an /æ/. We’re part of the “Mary/merry/marry” merger, so that may have something to do with it. That one also doesn’t get reduced to a schwa in rapid speech, so far as I can tell.
TSBG (post #25) mentions Carey McWilliams book “Southern California: An Island On The Land.” Well I do have a copy at hand, and have read it more than once. TSBG is correct. The book does indeed say it was a deliberate effort by the Anglo elite to de-Mexicanize the city’s image. And as a Loss An-gel-ees native I concur. I love my hometown!
There’s also a bit of doggerel about the different pronunciations, but I can only remember a snippet—something like “the train conductor said Loss Anjy-LEES…”—and can’t even remember where the punch line goes in concept. My Googling has been ˈfrütləs.
Caution: Conciliatory “plain English” pronunciation follows. 
Let’s all pronounce it “Lowe’s ANN jeelz” for a month, just to see how people react.