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?
I'm kind of uninformed about English ongoing changes, specially when it comes to NA dialects, so I hope that another poster chimes in.
That said, your description immediately reminds me two things.
One of them is /æ/ raising, a similar phenomenon affecting /æ/. Specially relevant here because /æ/ is right next to /ɛ/, raising /æ/ would leak it into /ɛ/ territory, and your description of how you pronounce bear hints that you're raising it. (Or alternatively you're lowering bare).
The second one would be the Northern Cities Vowel Shift, that likely applies to you, based on region. Usually the shift wouldn't apply to vowels before /ɹ/ (except /ɑ/), but perhaps this exception is going away.
Note that this type of sound change can actually split former homophones, specially verb vs. noun vs. adjective, given different stress patterns. For example, for plenty Australians the verb can (e.g. "you can do it!") is realised as [kʰæn]~[kʰɛn], while the noun can (e.g. "a can of beer") is [kʰæ:n]~[kʰɛ:n], with a longer vowel. While this specific split likely doesn't apply to you, the underlying mechanism might be the same.
Just for curiosity, how do you pronounce the verb "bear", as in "bear with me"? Is it homophone with "bare" or the animal?
I don't know anything about linguistics and can't add anything of value, but Northern Cities Vowel Shift is a great band name.