Q: Sync "Mark as Read" between iPhone and Mac

The fundamental answer is you need to consider POP versus IMAP email.  POP stores email on your client, so the mail on your Mac will be different from the mail on other machines.  IMAP stores the email on the email provider's server, so it will appear the same way on all your devices.

Regarding verizon.net, there is an additional complication.  Much (actually, I believe all) of the U.S. had Verizon email transferred to AOL several months ago.  So you would have to follow AOL's rules to determine if you use POP or IMAP.  I looked for those rules (at AOL.com) and could not find them.

Anyway, this post may be useful:

AOL VERIZON.NET EMAIL unable to set up as IMAP account in OSX (yosemite or el capitan)

I am not currently a Verizon customer.

But I think this website from Verizon will help you configure email.  Since I am not a customer, I could not try this site.

Again, remember to configure as an IMAP setup to gain the feature you are looking for.

https://www.verizon.com/foryourhome/VZRepair/vziha/Servicecad.aspx?ihaweb=FIX_AC TS

In the Mac's Mail app, you can use the menu to go to Mail > Preferences > Accounts > [your Verizon account] and look through everything there (under all tabs) to determine if you see something like POP or IMAP.  You should not see any POP settings for your server.

Well, you do not have to wait!

You can choose some other email service provider that provides IMAP.  Examples are gmail and iCloud mail.  You can then forward your Verizon mail to your new address.

That sounds like a good idea.

By the way, you can take the forwarding idea a step further if you wish.

Instead of informing all your contacts that you have a new email address, they don't even need to know.

You can forward your incoming messages.  And gmail, for one, allows you to do Settings > Accounts and Import > Send Mail As ... and use an email address you already own (your verizon.net address) as the "From" address!

I am 99.9% certain you can do this from iCloud as well.

Set up your iCloud email account.  Then from the Mail app, use the menu to go to Mail > Preferences > Accounts > [your iCloud account] > Alias (on the account information tab).  Edit the alias and add the one you would like.