Sorry to hear about your problem with the screw but it's not as big a deal as you might think. There are a couple of things that you can do.
1- You can take a needle and stick it under the screw head and then put pressure on it in a upwards kind of way while unscrewing it.
2-You can also get some glue that wont stick forever and glue your screwdriver tip to the screw and lift up while unscrewing it.
ie...modeling glue might work good.
Both of those ways should work if done the right way. Either way, just make sure that you take your time with it. After you get it out your going to find that you will still have a larger hole than the screw that came out.
I have read that some people have used a small drop of superglue in the hole to make the screw threads tight again. They say to put in a drop and then put the screw back in. Let it dry for a bit and then unscrew it and put another drop in the hole and screw it back down and let it dry again.
I think it might just work out fine for you. Just remember "DO NOT" use a bunch of glue. Just small drops.
Let us know if this works out for you & may God Bless.
KnotHole...