Yes, the snuffy nose was the bane of our newborn period. But I actually found the bulb syringe really helpful and used it along with q-tips. I don't use the one I got from the hospital, though. We really like the one that came with our baby care kit made by First Years. It's softer and the tip is tapered and smaller than the hospital-issue one.
It took a while to figure this out, but what works best for us is to put in a drop or two of saline solution, wait a minute if possible, and make sure the syringe is snug in the nostril, otherwise there is no "vacuum" to suck out the mucus. That's why the tapered tip makes a big difference. If the mucus doesn't come out all the way, we use a q-tip. Of course, this isn't terribly comfortable for the babe, but afterwards ds always looks much better when he can breathe freely again.