Using a new computer with Windows 10 and the strangest things are happening. I have saved documents that I use daily on the desktop and I have noticed that when I open one, another one pops up on the desktop and stays there until I close the one I am working on. It shows up with a dollar sign and a squiggly mark.
Any suggestions for me? This never happened before and I have been using this for years.

First, the desktop is a poor place to be saving documents. A much better location would be in your documents folder or a subfolder of that. You could then create a shortcut for that document and keep the shortcut on the desktop. (The desktop is a more fragile construct than is the documents folder.)

Second, if this is a document that you are constantly revising and then saving the revised version, a document is what you want. If you are opening the document, making changes, and using Save As to save it under a different name, you should be using a template instead. That is what templates are for.

What you are seeing is a locking file created by Word. It does this every time an existing document is opened. It is good that you see it. Some of your folder view options (like showing hidden and system files) must have changed.

I am saving the document under Documents and this is a short cut which I refer to many times daily. The interesting thing is, I just got a new pc with windows 10 and this never happened with the "copy" showing on my other computer..

If it is a shortcut on your desktop, no locking file should be created on the desktop.