Go to your local Lowe's store and find that fitting, Buy it, remove the new washer and install on old fitting. Also check that the fitting itself is not cracked. It will seep if it is and you will need to replace it. You can get the tool to compress the sleeve in their plumbing tools section. They should have all the plumbing parts used in these trailers.
Edit, I see you changed the washer, did you use one from the same fitting? If so then it is probably cracked, if it is the fitting you can cut off and replace, just need the tool. Also check the nipple it attaches to, if it is cracked, you will need to replace the faucet. As a plumber, i would look at that option as a opportunity to upgrade.
Dale, Karen, 2 Boys and the Girl
2007 Jay Feather 30U Some Mods
2006 F250 Lariat 6.0 Powerstroke Diesel with mods