Is there a way to export a non .ase file (something like a .gif or .png) and still retain the opacity settings of my layers?
I have these lovely clouds I’ve made on my world map but, when I export the file as a .gif or .png the layers merge and the opacity is reset to full. So my clouds, and cloud shadows, (both have their own layers) go back to solid white and solid black and it destroys the effect.
Or is there a way to “paint” it differently? This is maybe not the best way to phrase it but I hope it gets my point across.
What you’re looking for is not a common feature of image files, only a few formats support it, and none of those are display formats, and none of them are fully/widely supported by game engines or level editors. It sounds like you either need to save your stuff as multiple images with transparency (if you really need the different layers), or you need to save as a single-layer image with transparency preserved (if you just need the partial transparency but not the layers).
GIFs are unable to preserve partial transparency due to how they work.
The PNGs exported by Aseprite fully preserve partial transparency, so I don’t understand why you’d be seeing a white background if you didn’t add it yourself. It’s possible that the software you’re using to put your map together either doesn’t understand alpha channels, or (more likely) you’ve imported the image incorrectly. Some level editors have a “set transparent colour” option that replaces a given colour with transparency and treat all other colours as opaque, perhaps you accidentally used that.