Eliminating an Old App Store account

The App Store uses the account logged in at Settings ➜ Account ➜ Media & Purchases. You can simply log out of it from there, and you don't even need a password to do that.

Do note however, the Apps downloaded with that account will remain tied to it and require it to update, so you will need to find the apps downloaded with it and deletwe them from your device, and redownload them with your new account to stop any prompts asking for the old account when updating apps.

You have likely missed out mentioning a large piece of the story as to how you ended up in this place now as regards to the old account and the one you are using now. If you are somehow now using a new account on this device then the old one has not been fully removed and you will likely have future issues, potentially even being locked out of the device entirely. Ultimately if you want the old account gone then unless signing out of that account is the only step you need to do, you will instead have to get the old account working again. In order to properly remove an account from a device you have to be able to sign out fully and that requires that the account be working. Alternatively you can ask Apple to reset the device entirely and everything will be gone from it.

In the meantime, any apps downloaded or purchased with the old account will require the old account information to redownload them or to continue updating them. If you are using the device with a different account now, delete entirely any apps asking for the old account information and re-download them with the new account.