I'm afraid you will have to delete the apps (and potentially lose any data associated with them). The old Apple Account is built into them and it will always be required to update them or to redownload them. You can download the apps as new purchases with your new account (this is why Apple recommends you only ever use one account). If you had to pay for an app, you can try to get your old account working again so you can Family Share the apps without having to buy them again.
To learn how to set up Family Sharing, read this support article --> How to set up Family Sharing on your iPhone, iPad, or Mac - Apple Support
https://account.apple.com/faq
"How many Apple Accounts do I need?
Just one. Use the same Apple Account everywhere you sign in to ensure that all your Apple services and devices work together seamlessly and you can access your personal information from all your devices.."
P.S. You do not have to wipe your iPhone. You only have to delete the apps that are causing problems and then download them with your current account if you still want them.