Well, if she's sick, DO NOT let her cry or scream it out. She needs you now, more than ever. Try rocking her in a dark room with some soothing classical or lullabye music. Also, try 3-4 drops of Ocean saline in each nostril and let it sit for a minute before you try suctioning her nose out. It also depends on what kind of syringe you have---does it have a more narrow end that goes in her nose? That's what you need--if it doesn't, it just doesn't get up high enough to work. I got mine from the hospital when my son was born, you might want to ask your pediatrician where you can get one like that if you can't find it in the stores. As far the binky---I don't know that getting rid of it while she's sick is the way to go, but then again, maybe this is the optimum time?? (Nice answer, huh??) Here is a link to a book that helps you teach your children to get to sleep, get rid of the binkys, etc. I used if for both of my boys.
http://www.babycenter.com/0_the-ferber-method-demystified...