My problem is usually the opposite -- painful congestion rather than dripping or running. I take way too much Sudafed but nothing else works. I also use Flonase, saline irrigation, generic Zyrtec, etc., as well as an asthma inhaler, but only real pseudoephedrine relieves the painful congestion.
I have to spit a lot from coughing during asthma attacks. I try to wait until I'm not around anyone. Pretty easy on my rural rides, and much of the MUP. I don't worry about it as much when riding in traffic. If it gets a driver's attention away from their phones that's usually a good thing.