I will try what you said at some point. In the tutorial I was using (youtu.be/cC1A30mKf0U?t=414), on the left it showed a correctly distorted 2D map version of what he painted on the globe. If I used images instead of "texture painting", is that even possible? I could just have to get the continents the way I want and trace them with image painting.

you can directly paint on the sphere if this is what you mean, and you can paint on a separate layer that you put over a background, or several separate layers

You can also project an image with the Stencil option, in Texture Paint mode

maybe you should tell what is your final goal and someone will find the best method

Thank you so much for your help. I got your new method to work. What I want to do is arrange all the continents as I wish (I put the continent shapes centered on a 2x1 transparent canvas, since that seems like what the right size is to wrap a sphere, it wraps a 2x1 equirectangular picture of Earth perfectly onto a sphere). I can move the continent around and everything, but the end result I want is for those continents to appear correctly distorted on a 2D map.

You know how if you texture paint a vertical line from the top pole to to the bottom pole of the sphere in the 3D screen, on the 2D screen on the left (is that called the UV map?), it shows up as a distorted line that gets wider at the top and bottom? I want the PNGs of the continents I made to appear in 2D on the left and get similarly distorted as I move them around the 3D sphere. Somehow, the continent images have to be imprinted onto the surface of the base sphere...

(or as a last resort, I'll have to just trace them with the texture painting tool). I've experimented a bit with Stenciling, so I'll see if that works. It seems to require a perfectly square image and is inexact, because it just relies on the perspective and distance you're looking at the sphere with.

Ultimately the result only needs to be a guide to know how to stretch my shapes on the 2D map to be accurate, so the image doesn't need to be perfect.