:EDIT: - Here's the real cause of the issue!
The "transparent" color was solid blue, [0,255,0], with opacity set to 0. I noticed my "clear" material still had a tinge of blue, so I went back to the color and realized what's up. I'm assuming the color is showing as blue in the texture preview because Unreal isn't rendering the alpha channel in the preview.
-Original answer-
As @DMGregory hinted in the comment above, looking at the material that's using the texture made a big difference. I want the texture to be transparent, but it's showing up as blue/black when I view the texture AND when I view the material:
Changing the Blend Mode to Translucent seems to have correctly applied the transparencies, though. I still don't know why the TGA transparent values are showing up as blue, but if they're meant to be transparent then you need to have a translucent material.
Another problem I was having is that I've applied the same texture to two different materials, one that's supposed to be opaque and one that's supposed to be translucent. I would check the texture preview, see the blue, click the opaque material, also see the blue. It wasn't until I checked the translucent material that I saw it was showing transparency correctly.