"Here is the scenario. My company has a primary server and 2 DP's. Now the idea is to move/migrate over to new servers in the same domain and eventually, these new servers will move over to a new domain. "
Can't be done this way, at all. Once you build a primary in Domain A, you canNOT move the Primary to another domain.
"Now the new primary is not named the same name and is a part of the current domain. So my understanding is I cannot migrate SCCM this way. "
Correct, this won't work for your scenario. You cannot 'move' a Primary from Domain A to Domain B. It simply cannot be done.
"So I've been slowly building the new server manually."
Irrelevant if you've been building it manually. You still can't move this new primary server from Domain A to Domain B.
"Is there a way I can add all the users/machines to the new SCCM without affecting the current environment?"
Depends what you mean by "move", if you mean, you want all of those objects to be 100% managed by the new primary (even though its still a part of Domain A, and your end goal of having a primary server in Domain B will be impossible to achieve, sure, you can do that. I don't know why you would... but you could.
If you mean, "just discover the objects from Active Directory in Domain A, just so I can see some objects in the console, even though none of the clients will be reporting to this new server", sure you can. Again, I don't know why you would, since you can't move this new server from Domain A to Domain B. Ever.
"I have all discovery methods disabled and no client settings are deployed. I'm hoping to mirror the existing environment so I can test before flipping a switch down the road. Any suggestions?"
You will not be able to "flip a switch" to move this new primary, which is still in Domain A, to be in Domain B. Just plain can't be done.
My suggestion is that you will need to build this new primary in Domain B, the new domain. You can use the existing tools for Site migration (read the docs) for exporting things like your apps, etc., from your current primary in Domain A, to the new primary in Domain B (which will have no clients yet). Once those clients actually move to Domain B, ONE method to get those now-I-Belong-To-Domain-B computers to report to the primary in Domain B, would be that it's part of your migration script/plan, to re-install the client, telling it to SMSSITECODE=NEW RESETKEYINFORMATION=TRUE as part of the ccmsetup.