My third party email is dead. How do I change my Apple ID without access to my email

You are sign in here .  .  .  with your Apple ID & Password.

Your Apple ID is the personal account you use with all Apple products and services. It includes the email address and password you use to sign in, as well as the contact, payment, and security details that you use with Apple.

You can’t merge multiple Apple ID accounts.

See -> If you can't connect or sign in to iCloud

If you can't connect or sign in to iCloud - Apple Support

See this Apple article:

If your Apple ID is locked or disabled

If your Apple ID is locked or disabled - Apple Support (CA)

-> Get a verification code and sign in with two-factor authentication

Get a verification code and sign in with two-factor authentication - Apple Support

See this Apple article: Change your Apple ID

Change your Apple ID - Apple Support (SG)

Then see this Apple article: What to do after you change your Apple ID or password

What to do after you change your Apple ID or password - Apple Support

See -> How to remove Activation Lock

How to remove Activation Lock – Apple Support (UK)

If you need help removing Activation Lock and have proof of purchase documentation, you can start an Activation Lock support request.

You can also contact Apple support for help, -> Get Support (apple.com)

You have to show proof of ownership of your product, provide all documentation showing your purchase of the device.

If you do not have additional proof of ownership, contact the reseller. They may be able to provide the original purchase document.

Once Apple receive your proof of ownership, Apple will further review your request and provide an update.

My third party email is dead.

If you have deleted it, it is no longer valid, it simply no longer exists and it no longer works.

Contact the email provider of that 3rd party email.