This morning I started googling broken screws. Screw grades. All about the screw. And I am happy to report that, unless I am not understanding this correctly, that it isn't that I need a better more expensive screw, I just need to do a better job when working with hard wood. I found some useful information on a woodworking site:
To make screwing easier and prevent wood splitting and screw breakage, wax your screws before driving them. Drag the threads across a chunk of paraffin or an old candle to lubricate them. Avoid using soap, though; it can cause steel screws to rust.
They also talk about making the proper pilot holes, which I did do but now I see I could have done better. This is exciting, while I haven't been breaking screws very often, when I do it is usually close to catastrophic since the piece is usually mostly assembled and it's too late to turn back. If I cannot replace the wood because it is already affixed to the photograph or whatever object I've started working with I've had to adjust everything by working around it. Enter the screw extractor. This looks like a great way to remove the broken screw although the larger hole might be a problem if I have just a tiny area left to work with. I'm sure there are ways to plug the hole. Just thought I would share some useful information for anyone out there who also didn't know. It seems so rudimentary and I can't help but to think I should have known this, but the fact is, I'm learning it now and that is all that matters. Heck yesterday Mary B. left a comment asking if I used cold connection in my work. I had never heard of that term. I had to go look that up. My brain just doesn't think that way yet...I mean duh, a hot connection is welding and soldering.