Problem Configuring Log Shipping for DR Server Problem Statement - I want to configure DR site in SQL Server 2012 with the existing backups.

The backup scenario is as below.
FULL – on every Sunday, 
DIFF – daily except Sunday, 
LOG – every 15 minutes

Now, I want DR site for my production for which I am interested in using Log Shipping. But, the problem in Log Shipping is that it will initially restore full backup then regularly ships the transaction logs to the secondary server.

But, as in my present scenario if I use this then after every FULL or DIFF backup, my Log Shipping will break as the LSN will get change with respect to my current backup at secondary server.

So, is there any way to configure Log Shipping with using existing backups or do any one have any custom procs or scripts which will help us out?
I don't understand your issue.  You can definitely log ship while performing Full and Diff backups.  Are you trying to have some manually performed version of log shipping?
Performing a Full or Diff backup will cause no issues, log shipping just uses log backups so the chain will not be broken.

Have you actually tried this out, from your question it appears that you just think it will break.