But now it needs upgrade to 2012
I would suggest you to go with SQL Server 2014, unless you have a compelling reason to stick with SQL Server 2012.
AlwaysON is getting a lot of fixes back ported from SQL Server 2016.
we like to do AlwaysON but its mirroring behind AlwaysON so will it work or is there any other way ?
This is not true. The mechanism is different.
Mirroring will flush the log block to disk and once completed locally, it sends the block to secondary.
AlwaysON flushes the log block(s) in parallel. A secondary could have hardened log block(s) before the primary I/O completes. This design increases performance and narrows the NOT IN SYNC window(s).
total data is about 2 TB now that cluster is log ship to 4000 Miles another data center.
The data size matter to a certain extent - when doing an initial sync. What matters is the number of transactions occuring - meaning how busy your database is and the network bandwidth between the 2 data centers. The way you do index maintenance needs to be revisited as well.
There are other stuff like - backups (Full and log backups), application connection string chages to use multisubnet failvoer and hostTTL settings, reading from secondary, etc play crucial role when embracing a new technology.
I would suggest you to start reading : Stairway to AlwaysOn as a good start and then AlwaysOn Professional for some advance stuff.
Read : How It Works: Always On–When Is My Secondary Failover Ready?