Lots of short term blockings on SQL Server 2008-R2 database
This:

wait_type  wait_time_ms_per_sec
PAGELATCH_EX    733024.4
PAGELATCH_SH    328846.4
CPU_USED    7131.4
WRITELOG    6484.4


Shows that you have a latch contention problem, and only a latch contention problem.

Since


  @ below was run 15 mins after we did node failover


The CPU_USED row should be accurate.  And so your workload is only managing to use 7sec of CPU time per second.  This is like only running on 7 cores.  In the mean time you have 1000sec of page latch time, meaning that you have, on average 1000 sessions waiting to latch a page, and only 7 performing useful work. 

You are also running very old software on a very large server.  The throughput on this server is pretty high, suggesting that the operation it supports is important.  Also consider your username. So you should consider, strongly, engaging Microsoft Support or a partner with expertise in this sort of thing.

That said, the next step is to identify the source of your latch waits.  Generally these are either on a hot page of a table in your database, or a system page in TempDb.  The sys.dm_exec_requests.wait_resource should tell you which.

If tempdb, then Recommendations to reduce allocation contention in SQL Server tempdb.

If a table in your database, then post the DDL (including indexes) for the table and a the details of the queries that are blocking.  

There is a whitepaper on diagnosing and resolving latch contention on SQL 2008 here: Diagnosing and Resolving Latch Contention on SQL Server