There are a number of things you can do, once you cool down.
1) Carefully centre punch the broken screw, and drill into it with a new/sharp 1/16" bit. I assume the broken screw is about 3/4" long into the wood. Go into the screw about 3/8". Increase the size of the hole 1/32" at a time until you can get your smallest 'easy-out' started, and back out the broken screw.
or.....
2) Cut away the wood around the broken screw with a chisel until you can grip it with a pair of vise grip pliers, and turn the screw out. Then plug the hole, and cover it with an inlay, and put your new swivel screw in a different place, or drill out and plug the hole with a 3/6" dia. plug, re-drill for the screw (a little bigger this time), coat the screw with bees wax or bar soap to lubricate it, and turn it into the new hole. Cover the plug with a washer under the swivel screw, or a nice inlay.
No biggy.