"I have a runny nose / I have a running nose "Which is correct?

I have a runny nose 是正确的,意思是我流鼻涕了!
running 指水的流动,如:running water 表示流水;
而runny 是(指鼻子)流鼻涕的.

I have a runny nose

runny

adj.松软的, 水分过多的, 流鼻水的, 易流动的

I have a running nose

a running nose 意思是：流鼻涕的。
第二句对。