Disconnected the wires from the bottom of the silver metal box and tested for continuity between them - NO continuity. Not sure about, "With a meter check from each post to ground." Could you explain what this means?
I'm not sure how to answer that. It all depends on how well you like the range and how much time and money you want to put into it. We know the roaster thermostat is causing the short and the element needs to be repaired. Are the other elements and oven working correctly? You started checking everything when you came across the short but did you finish checking the rest of the range? If not, that's your starting point. Find everything that's not working. Keep in mind, parts for a 1947 Monarch electric range are no longer available. You can probably find a used part(s) and I believe the thermostat is rebuildable. If the elements and oven are working, you can use it as is without the roaster. If you want to get everything fixed and working it's no longer a repair, it's a project. I believe the range is rebuildable but it's going to take time and a lot of searching. eBay sells a lot of vintage parts. You're missing a few cosmetic parts like the light on top. I don't know how important that is.