Texture is super important. Is it throws you off i can understand. Though I'm sad because you don't get to have the flavour.

Though you could soak some feta in some dill pickle brine and have a feta burger with the taste of dills and avoid the snap you don't like. Oooor through some dills in a blender and then my with a condiment of choice

Lots of options! (Like just not having them is perfectly good too if that makes for a happier burger)