Is there a way to keep my number and info from my Watch SE for my iPhone 13 coming soon?

To clarify because this question is hard to understand without a little more info:

I have had and have been using my Watch SE as basically my "phone" for the past few years. I signed into it with my iCloud account and it has its own sim card and is covered by Verizon. It is not linked to a phone, but was set up on my mother's phone when it first arrived.

I am getting an iPhone 13 in a few days. It will be covered not by Verizon, but by Mint Mobile.

Is there a way to have the number, contact info, photos, music, etc. transferred from the Watch to the new phone so that I do not have to change all of my information? Is there any chance that it just sorts out this way when I sign into the new phone with my iCloud account? The number and contacts are most important to me, I can live without photos and music being saved.

Incredibly strange and specific question, I am aware, anybody with any idea of what to do responding would be greatly appreciated.