If I were you I'd just do the easiest thing for the both of you: don't bother with the nose drops! Apart from the messy mucus-faced look that he's sure to have if you don't wipe his nose all the time (which causes more chapping), having a runny nose isn't going to hurt him at all. I have never used nose drops for any of my children and they are completely unnecessary!
Cold meds aren't going to help him, you are right. I have been using different medications over the years and I've found most of them to give hardly any relief to my children. I did read about a study lately that said that children who took honey as opposed to children who took cough syrup did just as well and better in some cases with getting over their colds. I think this is mainly for "buckwheat" honey, which is very dark colored. I picked some up at Dillons just for this. They say that it is not known how well regular light honey that most people use would work in this situation. So try a drink of warm water mixed with honey...he's over a year now, so it's safe for him to take
I have heard that teething was blamed for runny noses when it's actually something else that causes the running, but who knows really? They say that it's just a coincidence. Science has come a long way but really it has so much farther to go, and no matter how good it gets, you and your child are going to be the only ones who will know what makes you both comfortable.
When my children have colds, I run a cool mist humidifier in their rooms, I use the pediatric vapor plug ins.....usually it has to be pretty bad if I do that though. The only time I actually mess with their noses is if they are too stuffy to breathe. The running is good because, better out than in, but if it's not coming out and they are too stuffy to breathe, "the solution to pollution is dilution". You take a nasal spray bottle (like the ones you have for allergy medication) and fill it with water. Give him a squirt up one nostril, let it sit a minute and then use a nasal suction thing suck out the congestion. Then do the same thing with the other nostril. You can do it without the added water for a really stuffy but very runny nose.
I hope this helps