The fact it broke off means the screw is probably very stuck. I would first use a penetrating oil (though WD-40 might also work), and let it sit for awhile, before trying anything further.
Hindsight is 20/20 of course, but it would have been good to do this first. If you see any corrosion or rust, or suspect it's needed and have time, this never hurts to do.
You may be able to use a small drill bit to extract the broken extraction bit, and then try again -- but be careful as you may damage the screw beyond my next tip..
If you can't get the tip out (or don't want to risk trying), I would probably use a dremel cutting wheel, and basically turn the screw into a slotted screw. This can be difficult -- not deep enough, and you'll just strip the screw worse trying to get it out, and too deep and you'll weaken the head so much it'll probably snap off when you try to turn it.
The next thing beyond this is to use a "reverse drill bit" or "left-handed bit". You can drill right into the middle of the screw itself (with your drill in reverse), and the bit should actually come out before you've gotten all the way into it. Use a bit that is smaller than the thread size, otherwise you'll strip the threads so you won't be able to put another screw back in. Again, judgment is necessary here, too small and it won't get enough grip, too big and you'll strip the threads -- though maybe by this point you don't care that much :)
![alt text](https://i.stack.imgur.com/asVqf.jpg)
It is highly likely that your beams are made of pressure treated wood. If so, you should try to use the same type of wood in repairing them.
SHIM - When there is a slight gap between a beam and a joist, that can often be cured with shims, slim pieces of wood (preferably of the same type, or if not available, a rot resistant type such as cedar). Shims should be forced in (tapped in with a hammer) so they are tight. Often you can drive shims from opposite sides. Since they are tapered, opposing shims can make a tight level gap filler.
LEVEL - If you cannot shim them adequately and need to move the joists to meet the beams, you need to make sure they remain reasonably level. They may have a slight pitch down away from the house, and that would be ok. You do not want to cause them to shift so that they slope toward the house. In some circumstances, that could lead to a drainage problem.
FILL - The holes in the beams can be filled with slivers of pressure treated wood that are coated with exterior glue. Force several narrow wedges in to pack the holes. Let the glue dry and then saw or chisel the surface flat. You can then drill new holes even close to the old ones.
Best Answer
Since the double ended screws are typically for attaching two wood surfaces together, start the screw in one of the pilot holes (just enough to stay put). Then align the other pilot hole and spin one or both pieces of wood together.
For example, if this is the below end cap on a wood pole, you'd put the screw on the end of the pole so that it doesn't fall and then spin the cap on until the screw is embedded in both surfaces and they are tight together. The only time you need to use pliers on the center piece is if you need to remove the screw.