Can I update a shared dropbox file while not changing the link?

Upload with the same filename and override the old one. I do this by simply dragging and dropping the new file (with the same exact filename as the old file) into dropbox. The same link gets preserved and I can even look at previous versions of that same file by right clicking.

The simplest thing for the future is to share the link to the folder instead of the file itself that way you can change the contents inside the folder without having to change links. Just have a folder per file.

You can share a folder containing only that one file.