CSS Module = Site Manager>Development CSS Stylesheets. I create a new stylesheet for the site and assign that in the Site Manager>Sites module. Then if I have another stylesheet say for instance a slider plugin, I'll create another stylesheet and add it to the page using the HEAD HTML webpart and adding this in the code
<link href="/CMSPages/GetResource.ashx?stylesheetname=MySiteSlider" type="text/css" rel="stylesheet"/> .
The stylesheet has something like this for a font import:
src: url('../../App_Themes/MySite/fonts/montserrat-bold-webfont.eot?#iefix') format('embedded-opentype'),
Here is an example of an image in CSS, again working without issue:
background: url("../../App_Themes/MySite/images/logoMobile.png") left top no-repeat;
An example of calling a custom js file:
On your load balancer, is it set to leave the user on the same server until their session ends? Or are they jumping back and forth between servers? I've set this particular setup to leave them on the one server until their session ends. Could be an issue, who knows.
What does your CSS and HTML look like? Maybe that needs to change. I'm almost positive its a problem with the load balancer setup.