After I renamed the $APPL_TOP/admin/log/$TWO_TASK/restart directory, the error changed to:
Adpatch error Unable to Flush Restart File in AIURWL()
After a lot of trial and error I noticed that there was a file in $APPL_TOP/admin/log/$TWO_TASK directory called adpatch_sessions_to_clean.txt. On a hunch I renamed this file as adpatch_sessions_to_clean.txt.old.
Voila, the patch proceeded without any issues and got applied too.
There is no reference of this file in Metalink anywhere. Inside the files were session numbers of some old sessions.