I was discussing this with my fiance, and she agreed with me in that she also speaks English in this manner.
I have found that, at least personally, I tend to speak several common homonyms in English in distinct ways: bear/bare, they're/there, where/ware. It's difficult to describe the differences in a concise way, but I'll do my best, and maybe use IPA where applicable, assuming I'm not using them incorrectly?
The traditional pronunciation of bare is [ˈbɛr]. I would completely agree with this, and while the dictionary might also say bear is pronounced this way, I would argue that I often hear it more as [ˈber] — a more closed sound with the lips pulled back in a smile. Sure, sometimes people will lazily say both in the same manner, but if I say [ˈber], the listener is going to recognize in a vacuum that I am speaking of the furry mammal, not the term to describe a naked person.
Similarly, there is rendered as [ðɚ]. There is a perfect rhyme with bare. I agree with this. However, they're is given the treatment of being a contraction of "they are", and it similarly has that closed sounded [e] instead of [ə].
Am I crazy, or does anyone else out there experience English this way?
That hints that you, your fiancée and others speaking the same variety are raising the vowel only in nouns, as nouns usually get a stronger prosodic stress. So it's a lot like the Aussie can/can split that I mentioned, except with another vowel.
I'm not sure but I think that they're listing homophones across multiple dialects spoken in USA. As such, for any given dialect, only some of those would apply. (It's for USA only for sure - note how there's no pair like more/maw or pander/panda).
Actually I feel like "can" is split here locally, too. "I can do it" comes out mostly like "k'n", it's highly reduced and closed. "A can of beans" is open, stressed. I guess it's a general feature of prosodic stress. I gotta read more about that
A bit of reduction is normal, since English relies heavily on the stress for the prosody. And by your description ("highly reduced and closed") you're likely realising it as [ə] or similar.
From what I've seen for your general area it isn't usually phonemic; the nearest of an /æ/ split would be Philadelphia and New York. Exceptions do happen though, and what I'm saying relies on studies from Labov from half a century ago.
A good way to test this out is to pretend that someone asked you to repeat the word, so you emphasise it, like:
If it's phonemic for you, even under emphasis you'll pronounce them differently.