The first thing to do is put the old URL into https://httpstatus.io. If it returns a status of 200, this means your web server successfully processed the request and it provided the requested page. So unless you are specifically telling Google not to index the page in the meta tags, .htaccess file, robots.txt file, or by some other method – then it will keep finding it and keep indexing it.
If Google keeps re-indexing an old page then the old page must still exist. Google cannot index something that does not exist.
What you want the old URL to return is a 404 (not found) status, if Google cannot find it then it cannot be indexed. Again, use the website above to confirm a 404 status once you have found and corrected the reason the page is still active. Often it isn't as simple as deleting a physical file from your web server and the issue is probably being caused by a badly configured URL rewriting setup.
Even better, if the old page has a new location then ideally the old URL should return a 301 (permanently moved) status along with the URL of the new page. This tells Google the page has moved and its index will be updated in due course. More importantly, any visitors managing to stumble across the old location will be automatically redirected to the new location.
Your web developer or server administrator should be able to help you with correcting this issue.