In Verizon Messaging app: When either you long-press new number from list of conversations and tap Add Contact in pop up; or you go into the conversation and tap the new number at the top to go to Recipient screen and tap new number at the top again, it brings you to Select Contacts screen where you can search and scroll existing, but gives no option to "Add New" anywhere.
T-Boner, I added a lot of my contacts to my phone from text messages I received. So I know the importance of being able to do that. I want to make sure you can add in a new contact when needed from Messages.
First, I would like you to close the Messages app. We're doing a simple reset by closing it like this; you won't lose any information from the app. In case you need the steps, here's how you can close a recently opened app: https://www.verizonwireless.com/support/knowledge-base-216718/
After you've done that, go ahead and open the Messages app. Once you have, go to your Conversations section, find the person you're trying to add as a contact. Hold down on their number till a new box opens. One of the options will be to Add to Contacts. It's important to note; the person cannot already be a contact. After you've hit Add to Contacts, a contact list will appear. At the top of that list is an icon of a person to the right of a + sign, and it says Create New Contact, click this.
When you select that choice, the option to create a new contact will appear, with the phone number pre-loaded. After that, simply fill in the required fields. Please give this simple app reset a try, and then use those steps to try entering the contact.