Carpenter glue would be great!! I'm pretty sure we jammed the whole toothpick in and then broke/rasped them off, though I'm sure breaking them in half would work fine (just more chance to get glue all over your fingers). Flat toothpicks are what we used too.
Drying overnight is always a good idea. The weatherweenies say both Saturday and Sunday are going to be beautiful, so you could easily make it a two-stage project.
I thought I'd put the toothpicks in on Saturday and replace the screw on Sunday. I've got some more shims around somewhere so I can give the door a little more support overnight.