To help with avoiding confusion for next time you have gotten your terms mixed up. The headset is the bearings and collars related to the bearings that allow you to turn the bars and fork for steering. The stem is the bit that goes into the fork's steer tube and locks in place and that you mount the handlebars into. It is the handlebars that you are trying to remove. And they only seem to be a single solid unit because of the amount of rust you have on them. They've corroded themselves together.

I'd suggest some penetrating oil and a pipe wrench. Because of the way the stem is made twisting the bars back from the top will aid in breaking the joint loose easier than trying to twist them forward and down.

I would also suggest that since you're trying to rebuild ratty old bikes that while you're at all this you should go ahead and remove the stem and the top couple of nuts on the headset and then take the forks out. This will allow you to clean, inspect and if, by some miracle given the condition of the rest of the bike, the bearings and races are still in good condition you can re-grease them and put it all back together.

How to do this and the tools to use you can see at www.parktool.com/repair . A web page that you should really get to know well. Hover the mouse over the part you want to work on and click when the right name shows up. In this case you want to select "Headset Service" and then Threaded Headset Service. But while you're there look at Headset Standards as well.