There is hope - don't despair.
The apps you are trying to update are tied to the "old" Apple ID, and will always be so. Your "old" Apple ID is still out there (unless, as Niel indicated, you called Apple and expressly asked them to delete it). Go to: Apple - My Apple ID, and click on the "Forgot ID or Password" link below the sign in section. Type in your old Apple ID in Apple ID box on the next page, and then follow the prompts.
If you do not have Two-Step Verification or Two Factor Authentication set up, then you should get to a screen that asks you to answer your Security Questions. If you don't remember the answers to all of them, then you will need to contact Apple Support to get a reset link sent to you: ACCOUNT SECURITY SUPPORT
Once you have your password reset, you can resume using your old Apple ID, or continue to use the new one if you wish. I have a few apps that I purchased with an older Apple ID, and the way I make the two ID thing as seamless as possible is by keeping the passwords to both IDs the same. That way, it doesn't matter which one is asking for a password.
Your other option is to delete the apps and re-purchase them, but since you have data you don't want to lose, I think the "recovery" of your old ID is the route you will want to take.
Best of luck,
GB
