Moving printers to consolidate servers without interrupting users [Windows 2003]

Greetings! 

How do I go about moving network printers served from ServerA to ServerB, decommission/shutdown ServerA, and be completely transparent to users without causing them the need to remap printers?

My company is still at Active Directory level Windows 2003 SP2, and has all of our printers on several Windows 2003 SP2 servers. I am in the midst to decommission several old servers and consolidate print servers and then upgrade the domain to Server 2012 levels.

Right now, a user may have a network printer setup so that it is listed as "Printer1 on ServerA"

I want it so that when I move the printers to ServerB, users can still use their already setup "Printer1 on ServerA", instead of having to map them with "Printer1 on ServerB". Although, any new printers added from the consolidation point forward to a users PC would be "Printer1 on ServerB".

My plan is to use my Windows 7 PC to:

1. Use Print Management to migrate/export ServerA printers to a file.
2. Use Print Management to migrate/import that ServerA printers file into ServerB. Using "Keep existing printers" in Import mode and "List All Printers" for List in the directory. [That would add ServerA's printers to ServerB while keeping ServerB's printers. But I'm a little unclear which option to use for List in the directory.]
3. Shut down ServerA.
4. In our AD DNS, delete ServerA's A record and then either (a) add an A record for ServerA using ServerB's IP address, or (b) add a CN record for ServerA to point to ServerB's FQDN. [I think (a) is the better option, right?]
5. And then after the DNS refresh, the jobs should just come out of Printer1 on ServerA (as already listed/setup on my workstation) even though Printer1 now resides on ServerB.

I tried that plan last night, and things just didn't quite work out. My workstation said Printer1 on ServerA was Offline and I couldn't print to Printer1 on ServerA (which was imported onto ServerB) even though my DNS had ServerA with ServerB's IP.

For now I brought ServerA back online and people are working.

What am I missing or should be doing differently? Any issue with local security that would prevent one server to not accept commands directed to another servername? Thanks for your input!

Visual nutshell:

Printer1/ServerA ---> shutdown ServerA ---> Printer1/ServerB while those with Printer1/ServerA definitions on their workstations would still print.