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?
Nope, bear and bare are the same like hair and hare. There, their, and they're are also the same. What you are talking about are dialects, there are many in the US. I've heard Columbus, OH is the most pure dialect free English.
OP is describing differences of pronunciation. You're prescribing that they "should" be pronounced the same (why?). Those are two different cans of worms.
Given OP's usage of IPA odds are that he's fully aware of that. Also note that a dialect can be defined from a difference of pronunciations, so "what you're talking about are dialects" sounds like correcting what's already correct.
No such thing as "dialect free". Except maybe for languages spoken by really small and close-knit communities; but certainly not for anything spoken natively by lots of speakers, like English.