I have completed my B.A (HISTORY) undergraduate from TAMILNADU OPEN UNIVERSITY and am I eligible to puruse LLB In full time

You can check this  for particular University law admission rule.

In my view you can pursue 3 year LL.B from any college other than NLU 5 year course.

Bar council allow open university degree for LL.B pursue but must be from an approved university  ( ODL/distance approved )