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?
It's really common for borders between dialects (and languages, too!) to not coincide with any sort of political border. And to make it even messier, sometimes there's no border to begin with, only gradual transitions - and I think that's the case here.
Based on this map your dialect would be probably Midland English, but you live so close to the Great Lakes English-speaking region that you're actually speaking a transitional dialect.
Looking the map over, I am actually in the Fort Wayne/South Bend region and it appears that this is transitional began Midland and Great Lakes.
Thinking about it, it may also be useful to note that there is a strong Pennsylvanian Dutch influence in my area given a large Amish population. I do not speak it but I can't imagine that there hasn't been at least a little bleed into the English here.