Does this sound like Pink Eye in my Cat?

My poor Kitty woke up this morning with a sore eye.
The third eye lid is showing and all white and seems to be puffy, she keeps pawing her eye, it’s not red as such but she has some dripping from it.
Does this sound like conjunctivitis?
I have just adopted her so she has had all her shots.
Also how do I treat this without getting it myself?
She licks all the time so can you get it through saliva or is it only if i touch her eye and then mine?
sorry a lot of questions, thanks for your help.