I looked at your other question yesterday, went to your site, after a menu click the header would indeed stop scrolling. There's a ton of scripts on there from all sorts of frameworks, themes and Google. Which one is not working? Or which one is supposed to do whatever it is that gets broken?
I checked again, in the console there are a couple of errors and the links aren't working like yesterday. Are you sure you need to relinquish the $? jQuery and jQuery mobile should not have an issue with eachother using $. Other libraries might, but jQuery is pretty common these days.
The fact that you did relinquish the $ might even be why other libraries (that require jQuery) stopped working.
One of the errors in the console is that jQuery is not defined in:
I would yes, if you click a link in the menu the console will show:
TypeError: $ is not a function
$( "#mypanel" ).trigger( "updatelayout" );
So you have parts relying on the $. Was the problem before only with the non-fixed header? That seems like a less worse situation, and if the rest of the jQuery dependant parts kept running at that time it could just be a problem unrelated to jQuery/jQuery Mobile.
I'm still getting the three typerrors on the homepage, but clicking a link works and the header then stops resizing. If I however go to that page directly, the header does resize so it's not something on that page specifically.
How do you fetch a new page exactly? Through Ajax it seems to me. It could be that you are retrieving the entire new page, but then not binding the resize listener again to the new header perhaps.