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?
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.