Any equirectangular projection is going to be infinitely distorted at the poles which makes it extremely difficult to complete the map just from that format. Drawing them on a 3d object and then unwrapping the UV is often the best option, but not everyone has access to those skills. Your best bet is to draw each continent as a regional map and then after that paste them into a world map. For the poles I think Gprojector can take your region and you can move it up to the pole and get the right distortion to project it in equirectangular (though frankly there are better options for world map projections).
Here is a step by step for Blender:
-Open Blender up, press new scene then press x to delete the cube
-press shift a, select mesh and then select UV sphere
-press tab, this should put you into edit mode, you'll see the nodes and connection lines in black
-(you can move the mouse to the empty space in the window with the sphere in it press the middle mouse button and hold to move your view around to navigate in the 3d space)
-you want to switch your select mode, press 2 to make it edge select (or in my layout there is a button in the upper left hand section of screen you can use)
BlenderWorldTutEdgeselect.jpg
-Move you cursor over a vertical line in the sphere and press alt left click
BlenderWorldTutCursorPlacement.jpg
-One that edge is selected and highlighted as shown press ctrl e to bring up the edge menu scroll down and choose "mark seam" it should turn red (and yellow since it's still selected)
-extra step if you want less distortion on the UV map:
-Press alt a to clear selections
-move your cursor over one of the horizontal lines surrounding the poles and press alt click it should select the ring
BlenderWorldTutEdgeselectExtra.jpg
-navigate to the bottom part of the sphere by moving out to an empty area and middle click and hold
-move the cursor over the same line on the bottom and press shift alt click, both rings should now be selected
-press ctrl e and select "mark seams"
-end of extra step
-Now press a, everything should become selected (if not press alt a to deselect everything and try again)
-press u and select "unwrap"
And then this tutorial should help you out to paint on the globe. Afterwords you can export the texture as a map (it'll have a kind of strange projection though, still you can warp and edit it as needed in a raster program if you want to make it work in a different projection.
There are obviously a lot more advanced ways and tricks that help with this process, but this could get you started.