Outstanding Support - THANKS!
Let me go into a bit more detail here, since the solution may help someone else with the same problem.
In DW, when setting up your servers, using a LINUX host (I'm using HostGator), the FTP address is your site URL, the "Web URL" is your site URL prefixed by http:// -- there should be nothing after the top-level domain extension (.com or .org or .us or whatever).
The root directory should be public_html
I had all that just fine on the remote server. However, on my testing server, in response to the 404 error message that said the testing server should map to (URL)/public_html/(etc.), I had added /public_html to the Web URL field, and it should not have been there.
As is usually the case, this was something small that I overlooked, and I'm hugely grateful to Ray for taking the time to point this out to me.
This is the kind of support that (a) I hate to ask for because it takes up people's valuable time - but I was down to grasping at any help I could get; but also (b) will keep me, and many others, I should think, a loyal customer of Web Assist.
Thanks again for the help.