What you said it's fine. You can also use: Tengo mocos, estoy constipado (informal) and tengo mucosidad nasal (too formal).

Oh, I agree there, Fer!

Tengo flujo nasal.

I agree with the rest of the answers, but I'd say that "tengo mocos" is the most commonly used expresion by far.

Hello there! If I may join in, perhaps I'd say "Me gotea la nariz", which quite literally refers to mucus running from your nose. I hope this helps!