If I'm stuck in the same area as someone who is clearly sick (runny nose, coughing etc) is there some combo of short/long breaths or nose/mouth breathing that's a better defense against catching it?

I think a mask is your best option. If you’re trying to prevent against other people’s illness, you probably need an N95 mask, something that seals around your mouth and nose. A surgical type mask is more to control the droplets and spread of illness coming from you.