I'm not entirely following.
When you create a new document, the background layer is always going to be filled with white. That's normal, and I am not sure that there's any other way to open a new document. Are you saying that you copy parts of an image layer to a new layer? Is that not working for you?
It might be easier if you try to duplicate this problem, and then copy your entire workspace to the clipboard, and paste it to this thread (Ctrl v)
Sorry I wrote 'white' instead of 'transparent' in my sentence ("background always used to be transparent"). What I meant is that when I used to make a selection with the selection tool, make it a new layer and turn off visibility on all other layers, the background of the selection used to be transparent and not white.
Without system information and other details like exact version of PS, tool settings, document settings etc. nobody can tell you much. From display issues to a wrong setting on a tool that simply fills empty regions with the background color this could be anything.
We still need that screen shot. It is super easy to do. If you have a Print SCRN key, press that to load the desktop to the clipboard, and paste here with Ctrl v. Then we at least have a chance of seeing what is going on.