If I sign out of my new Apple ID into my old one, can I retrieve my old stuff?

You can sign in and out of Apple IDs without any issue.

When you sign out of Apple ID then all the services associated with the Apple ID will also go away at the same time when you sign in with an Apple ID all the services associated with the Apple ID will flow back

If you use an @icloud.com, @mac.com, or @me.com email address as your Apple ID, you can’t change it to a third party email address. But, you can change it to an alias of your Apple ID address. For example, if j.appleseed@icloud.com is your Apple ID and you also have j.appleseed@me.com on file in your account, you can change your Apple ID to j.appleseed@me.com. When you change your email address, the Apple ID that people see when they share with you also changes.

• To find an eligible email address that you can use as your Apple ID login, sign in to your Apple ID account page and click Edit under the Account section. You see Reachable At and a login alias with a question mark icon . If you hover over the icon, you see a message that says, "This is an alias of your Apple ID email address."

Sign out of all Apple services and devices that use your Apple ID. If you want to use your iPhone, iPad, or iPod touch to change your Apple ID, stay signed in with your Apple ID on only that device.

Change your Apple ID - Apple Support

https://support.apple.com/en-us/HT202667

You may get better and prompt responses if you may kindly provide the iPhone Model that you have and the iOS version if it is known to you.