The cough suppressing part of Nyquil is Dextromethorphan aka DXM (Robitussin DM). The sleepymaking part of Nyquil is Doxylamine, a sedating antihistamine. The main reason you shouldn't drink while taking Nyquil is that it has Acetaminophen (Tylenol) in it, and alcohol + Tylenol is hell on your liver.
If you only want to suppress your cough, you should be taking just DXM by itself, either as Robitussin DM syrup or in one of their newer (easier to swallow) cough gels. Robitussin DM also has 100 mg of Guaifenesin, which is an expectorant. You can buy Guaif by itself either in syrup or capsules, that's what Mucinex is, for example. It helps thin your mucus so you can have a more productive cough - gross I know :crazy:
But personally, I just take oil of oregano for respiratory stuff. The tincture is great, very potent, but extremely pricey at $75 a bottle or so. But that's like a concentrate which lasts me over a year, you mix one drop per teaspoon of olive oil. You can get capsules for much cheaper.
Try it, it's great! Much more potent than echinacea and zinc (which used to be my standbys before I found oil of oregano) and especially helpful for coughs. Google Oregacillin and see for yourself - that particular brand of wild oregano supplement has been researched and tested.