Tanins, especially in black tea, help with the infection and swelling. Yes, please use cold wet teabags. Just get comfortable, close your eyes, and let them do their work.
Used teabags are ok. This is why it's good to keep a box of cheap generic teabags in the back of the cupboard.
If all you have is good tea, use a paper Teasack to make your own teabag, enjoy a nice cup then chill the homemade bag for an hour or so before placing on eyelid.
Once you've used a teabag on your infected eye, throw it away so that you don't re-infect the eye with a subsequent application.
The pinkeye cure my family used was salt water: dissolve a teaspoon of table salt in a glass of warm (not hot) water, and let it cool to lukewarm. Fill a large spoon with this homemade saline solution, lean your head forward (like you're looking down at your feet), and hold the salt water on your open eye for several seconds. Get a clean spoon and do it again.
Can be repeated every couple of hours.
I know it sounds gross (and to some impossible) but salt was given to us as an amazing curative, and these old-timey cures have been used for generations.