What happens when you get an Esior's Haven badge I've checked everything in the town, but there's nothing you can purchase with these badges and I've never gotten one before do you obtain one every time you fill the trust bar?

To be honest, I am not sure what they are used for. I don't have any, and I don't see anything that uses them, so they may be used in the quest line itself.

In the Sanctuary of Saerielle, you can use badges you get there to buy avatar items and potions (and maybe other things not really sure), but I can't answer about Esior's Haven badges.

As far as I can tell, the badge system was never actually implemented for Eisor's Haven. I'm not sure how you would even get a badge for Eisor's Haven; there's certainly nowhere to use them and I'm pretty sure they're not a part of the game.