Iphone keep asking Apple ID Verification

You should be able to delete the apps. If necessary, check your purchases history to see which apps/purchases weren’t made by the current ID. 

Purchase History Review.   

You do not need to format the phone.

If you are locked out of your Apple ID account you can go to https://iforgot.apple.com

This will provide you with the recovery steps you put in place for this situation.

Here are some support articles that should help:

If you forgot your Apple ID password - Apple Support

If you forgot the answers to your Apple ID security questions - Apple Support

If your Apple ID has been locked or disabled – Apple Support

This is likely to be your problem. See What to do when the Mac App Store shows a different Apple ID for software updates https://www.macworld.com/article/3267709/mac-app-store-apple-id.html. This document refers to a prior owner, but is also applies to cases where a current owner's old Apple ID is the problem.

Basically, If an app is purchased/downloaded using one Apple ID, then you can't update it with with another Apple ID. The app is inextricably linked to the first Apple ID. You'll have to delete it and repurchase/download it again with the second Apple ID unless you use family sharing (see below).

As an alternative to deletion, you can set up family sharing between the IDs to share purchases made with them. See:

Family Sharing  https://support.apple.com/en-us/HT201060

Set up Family Sharing https://support.apple.com/en-us/HT201088

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But, activation lock also could be your problem. See Turn off Activation Lock https://support.apple.com/en-us/HT201441. 

If you are the original owner: If you have iOS 11 or later and your account is protected with two-factor authentication, use get help with a forgotten passcode or disabled device to recover your passcode. Otherwise, use forgot your Apple ID or your Apple ID password to recover the needed credential(s). If you can’t recover the needed credential(s), contact Apple Support and present your original purchase receipt for your device from an authorized Apple seller with the device’s serial number on it. You’ll need a photo ID too. If you can do neither, then the device is useless to you.

If you acquired your device second hand, you usually will need to obtain the needed password from the prior owner or have the prior owner sign out of iCloud; erase the device; and remove it from their device list. If unsuccessful, in the rare case your device came activation locked from an authorized Apple seller, then contact Apple Support and present your original purchase receipt for your device with the device’s serial number on it. You’ll need a photo ID too. If you can do none of the above, then the device is useless to you.

In either case, you can use the suggested support options below if you have the needed receipt. Alternatively, if an Apple Store is open in your area, then you can go there to deal with your problem. In this case, you can make a genius-bar appointment (https://www.apple.com/retail/geniusbar/) if you want.

Get a call from Apple support to help solve your problem using this

link https://getsupport.apple.com/?caller=kbase&PGF=PGF63005&category_id=SC0245&symptom_id=23362  Be patient. Sometimes Apple phone support is not available.

Apple ID Support https://support.apple.com/apple-id

Apple Support Twitter  https://twitter.com/AppleSupport  There may be an hour or so wait.

Apple ID account security for country-specific andregion-specific help https://support.apple.com/en-us/HT204169

Get Support: We’re here to help https://getsupport.apple.com/. Go through the options to find the best support option.