embluk

Stick all of the code into something like JSFiddle so that we can take a better look.

Yes the next() method will select the next sibling element (next element at same level in the DOM tree)
if you pass a parameter like the class name selector it will only select the element if the class matches.
https://api.jquery.com/next/

This maybe:
$(".linkDropDownButton").on("click" ,function() { $(this).next(".sectionDropDown").slideToggle(); });

Looking at the code it looks like you want to attach the click event to ".linkSectionStyle" not ".sortable"
$(".linkSectionStyle").click(function(e) { $(this).find(".sectionDropDown").slideToggle(); }); EDIT: I was typing this while you found the same answer your self. Glad you figured it out

$(".sortable").click(function(e) { $(this).find(".sectionDropDown").slideToggle(); }); ...should do it
 
EDIT: Fiddle here: https://jsfiddle.net/Nillervision/uagwLh7a/

Perhaps this could help - http://jonrohan.codes/fieldnotes/vertically-center-clipped-image/?