Migrating exchange 2010 to exchange 2010 on new hardware - Need assistance moving public database

Option 1. If you dont have time I can you export the data to PST, Delete the current PF and create a new PF in the newly installed Exchange and import the PST. Dont forget to tag these PF in your database and OAB distribution.
Use admin account to import https://support.microsoft.com/en-us/kb/287070 

Option 2. You can use MoveAllReplicas.ps1 script
https://technet.microsoft.com/en-us/library/bb331970.aspx

DB Migration has begun I will post back if more help is needed thank you

Hi
After Exchange Database migration you can easily move public folders to new exchange server of same or other version.
Go through below link where you can find three different way of Public Folder migration.

http://exchange-server-guide.blogspot.in/2015/12/move-public-folders-from-2003-to-2010.html

Hope It would be beneficial for you.

The Public Folders replicas will take time to populate in the new server, so, you can also try to export the content into .PST, add replica and then import the .PST  back.

However, to move all public folders in a public folder database on one server to a public folder database on another server, use the MoveAllReplicas.ps1 script. This script replaces a server with a new server in the replication list for all public folders, including system folders. To move all the replicas of a public folder subtree from one server to another server, use the ReplaceReplicaOnPFRecursive.ps1 script. This script adds a new server to the replication list for a public folder and all the folders within that hierarchy.

Please have a look at this blog: http://sharadtech.blogspot.com/2011/11/hot-to-move-public-folder-from-one.html

Move Public Folder database from One server to Another: https://msexchange.me/2014/04/28/move-public-folder-database-from-one-server-to-another/

Hope this helps!