Will my apps stay on another Apple ID?

First, Son 1 and Son 2 will need their own unique AppleIDs - these being different to your own AppleID.

Golden Rule: Don’t share an AppleID between users. Each user should have their own AppleID for their own exclusive use - linked to their own primary email (and secondary/recovery email addresses and trusted telephone numbers). 

Having ensured that you have separated your identities, as the family organiser, you can selectively share purchased Apple Services, Subscriptions, Apps and Media using Apple’s Family Sharing features:

Family Sharing - Apple

Set up Family Sharing – Apple Support

Share App Store, iTunes Store and Apple Books purchases with Family Sharing – Apple Support

How to create a new Apple ID – Apple Support

https://www.imore.com/how-set-up-family-sharing-iphone-ipad

So as to avoid unintended sharing and merging of your private/sensitive data (including Passwords) with your family, instead of trying to sign-out from a device that you intend to repurpose or give-away, it is essential that the device is properly prepared - and setup from scratch by the new owner (Son 1/2). To prepare the iPad for reuse, follow this procedure:

What to do before you sell, give away, or trade in your iPhone or iPad - Apple Support

Proper preparation, as described, will avoid many other potential future complications and difficulties that might otherwise occur. After setting-up Family Sharing, you’ll be able to install your existing purchased Apps on devices that are within the family sharing group.

You may find this information to be helpful with setting-up an AppleID for a minor - and considering appropriate controls:

Apple for Kids - Official Apple Support

Help your child set up an iPhone, iPad, or iPod touch - Apple Support

Family Sharing and Apple ID for your child – Apple Support

Use parental controls on your child's iPhone, iPad and iPod touch – Apple Support

First, Son 1 and Son 2 will need their own AppleID - this being a different to your own AppleID.

Golden Rule: Don’t share an AppleID between users. Each user should have their own AppleID for their own exclusive use - linked to their own primary email (and secondary/recovery email addresses and trusted telephone numbers). 

Having ensured that you have separated your identities, as the family organiser, you can selectively share purchased Apple Services, Subscriptions, Apps and Media using Apple’s Family Sharing features:

Family Sharing - Apple

Set up Family Sharing – Apple Support

Share App Store, iTunes Store and Apple Books purchases with Family Sharing – Apple Support

How to create a new Apple ID – Apple Support

https://www.imore.com/how-set-up-family-sharing-iphone-ipad

So as to avoid unintended sharing and merging of your private/sensitive data (including Passwords) with your family, instead of trying to sign-out from a device that you intend to repurpose or give-away, it is essential that the device is properly prepared - and setup from scratch by the new owner (Son 1/2). To prepare the iPad for reuse, follow this procedure:

What to do before you sell, give away, or trade in your iPhone or iPad - Apple Support

Proper preparation, as described, will avoid many other potential future complications and difficulties that might otherwise occur. After setting-up Family Sharing, you’ll be able to install your existing purchased Apps on devices that are within the family sharing group.

You may find this information to be helpful with setting-up an AppleID for a minor - and considering appropriate controls:

Apple for Kids - Official Apple Support

Help your child set up an iPhone, iPad, or iPod touch - Apple Support

Family Sharing and Apple ID for your child – Apple Support

Use parental controls on your child's iPhone, iPad and iPod touch – Apple Support

Apps bought on one Apple ID are tied to that ID. They won’t be available on a different ID (aside from being re-purchased). They will be available on any device signed into with the same ID they were purchased on.

Yo are perhaps missing an essential and critical point. Your own AppleID is known to you - and ONLY you. Your AppleID shouldn’t be used with either of your Son’s devices.

They sign-in to their devices and Apple services (including the Apple App Store) with their respective AppleIDs - and can access all the services and Apps that YOU have purchased and shared (via family sharing) with family members. All your past App purchases will be available to all member of the family sharing group.