Or, rather, JEI says it can use cool oil to make fuel, but whenever you try to put oil into the Distillery, it goes into the output tank (???)

Version 1.0.0 of Gadgetry
Version 2772 of Forge
Version 7.99.18 of Buildcraft

Maybe this one helps you: https://minecraft.curseforge.com/projects/what-fluid-fix

No. That mod when you have two different kinds of the same fluid, and forge doesn't bake the texture for one.

My issue here is that, with Buildcraft installed, Gadgetry digs up Buildcraft oil, and Buildcraft oil cannot be processed in the Gadgetry refinery.

That being said, I just realized I left out a crucial detail for this issue:

I also have Tech Reborn, which also registers an oil fluid.

That was my thought exactly. But I didn't know that this mod is only for the visuals