Retrieving contacts via old account to new one
You may retrieve them by doing the following. 


Open any web browser, and go to icloud.com on your computer.
Log in with your iCloud account (Your Apple ID and password).
Click on Contacts. Click the gear shaped Settings button on the bottom left, and click Select All.
Click the gear button again, and click Export vCard… to download contacts from iCloud.


Then, you reupload by signing in with the other Apple ID following these steps. 


In iCloud Contacts, click the Action pop-up menu  in the sidebar, then choose Import vCard.
Select a vCard to import.