YxxYYYY,
You invariably saw my screenshots above that I posted to Gail showing how the Amazon App does not display all the options available unless one clicks on > and then selects from the menu that ensues. Have you tried that? Let's say on Pinterest since you are sure it had badges... It is always possible that the App developer changed the App in an update, worth checking perhaps. Do you have another iOS device like an iPad where you could check it? Or with a friend? I am trying to isolate the issue to just your phone or not....
If your iPad shows badges on Pinterest then we know it is an issue with your phone, if it does not then the issue is with Pinterest an not much you can do from your end other than contact them.
If the issue is your phone, two options come to mind, reinstall the app but you already did that, and then do a complete backup, reset and restore as per the instructions below. It will take you 30-60 minutes. If you are going to do the restore, first import all you photos to your computer and also any important files or data. Then delete the Pinterest App and all others that may be misbehaving (that way when you restore the phone you will not bring them back). Then after the full restore install Pinterest again and all others and check. That is about all I can think of to do if all your settings are already configured as they should. Good luck.....
Use iTunes to restore your iPhone, iPad, or iPod to factory settings - Apple Support