Osmania University

Yes, OU offers B.Sc Psychology for Andhra Pradesh students. You can apply for admission. If you meet the eligibility criteria then you will definitely get admission there.
All the best.

Part-A (40 Marks): 1. Business Laws (10 Marks
Financial Accounting (10 Marks
Corporate Accounting (10 Marks
Auditing (5 Marks
Fundamentals of Information Technology (5 Marks):
Part-B (60 Marks): Business Economics (10 MarksBusiness organisation And Management (10 Marks
Business Statistics (10 Marks
Financial Services-Banking and Insurance (10 Marks)
General English (Comprehension) (10Marks)
General knowledge and current affairs (10 Marks). .

Check out the courses of Osmania University through this link :
https://www.shiksha.com/university/osmania-university-hyderabad-25100

Osmania University MCA Eligibility : Should have passed Bachelors Degree examination of Osmania University or any other University recognized by Osmania University with minimum three years duration with at least 50% marks ( 45% marks in case of SC / ST / BC categories ) in the qualifying examination with Mathematics at 10+2 level or at graduation level.

No, you can apply in final year.

just check out this link
https://www.shiksha.com/science/course/m-sc-in-biotechnology-osmania-university-college-for-women-kachiguda-hyderabad-287076

The eligibility says that you need to have 40% marks in Part II with Psychology, as an optional with BA/BMR, M.Ed/ MA (Education)/ M. Sc (Home Science)/ Diploma in Child Psychology or Industrial Psychology.