Hi,
You should redirect the old url's to the new ones using a 301 redirect. Normally it's not really necessary to submit for re-indexing, Google is quite good in finding new url's itself.
The fact that Screamingfrog still finds the old url's is a sign that on your site you are still using links to the old url - you should update the internal links to the new url (you can easily get this info from Screaming frog - take the url's that have a 404 - check at the bottom of the page "In Links" - this will give you all the pages where the links need to be updated.
rgds,
Dirk