Is there a correct way to change SharePoint web applications / IIS
Sites to a new application pool?

I created a new app pool at the recommendation of MS support to fix a
problem we are having with SharePoint, but I cannot seem to get the
whole site to recognize the new app pool. I went through IIS manager
and changed the app pool on the websites that were needed, but it
left
behind _layouts/images and _layouts/inc. I can manually move the
images
as well, but not the inc. There is no _layouts/inc on the server.

The other problem I have is that if I do anything to the web
application through central admin after switching application pools,
it
defaults the IIS sites back to the application pool originally
created.

Is there something I am missing? Is there somewhere in central admin
that I cannot find to do it?

Any help on correctly changing the application pool of the web
application is appreciated.