That said, is it possible to be "fluent" in both qwerty and dvorak at the same time? If not, are there any good solutions to the situation I found myself in?
I've switched to Dvorak a few years ago and I've been fluent in qwerty and dvorak most of the time since (only slightly slower in qwerty than I used to be; much faster than I used to be when I use dvorak instead).
I found that the only part where switching back and forth was really hard was the beginning: I needed a few months until dvorak felt natural enough so that switching back and forth wouldn't confuse me (and hurt the learning curve). After a few months, switching back and forth was a little awkward, but quickly became entirely natural after I got used to it (school computers and a few games didn't easily let me change to dvorak, so it was nice to be able to work either way). So I think if you practice this a bit, it should be fine :)
Incidentally, once I was comfortable with both layouts, learning other layouts was extremely easy in comparison to learning dvorak. It took me only a few hours until I felt more comfortable typing Japanese characters (Hiragana direct input) than I did spelling them out, even though I had to learn it by hitting all keys and seeing what produced which character. It felt a bit like it did with languages -- once you knew two well, learning other similar languages is a lot easier than it was for the second :)