The newspaper and paste technique works wonderfully and is exactly what your mom used to do. I made several like that when my nephews were little. The key is making the paste properly: you boil water with flour (like a thin paste, no lumps), let it cool and you're done. Some people add a bit of salt so it lasts longer, which makes sense if it's going to be outside for a while before the party.
To build the structure you use a big inflated balloon as the base, tape it to thick cardboard so it doesn't move. Then you tear newspaper into strips that aren't too wide, dip each one in the paste and wrap them around the balloon, overlapping them like roof tiles. You need quite a few layers, at least 8 or 10, so it comes out sturdy. While it dries between layers (ideally wait a day or two), you can make the face, ears or whatever with cardboard or papier-mâché. Once it's really hard, you mark where the hole goes for the candy, cut it carefully, and it's ready to decorate with paint or colored paper.
That thing about paste coming off with moisture is true, so if they're going to play outside and it's really sunny or humid, it's better to stick the final paper on with homemade paste but make sure the bottom layers are completely dry. If you want it to be super sturdy, you can make the last layer with thicker paper.