Hi,
I worked in oncampus job in university under F1 visa for the first 4 months of 2008 and I am having a W2 for it which is for Texas state. In Oct 2008 I transferred to H1B Visa and started working for a company in Ohio from where I got another W2.

To be simple, two W2's from different states. First W2 for oncampus job under F1 Visa and Second W2 under H1B visa.

Since I did not accumulate 183 days under H1B , Is it enough to file the 1040NR and 8843 which I used to do usually for W2 under F1 visa even though now I am under H1B?

Please help me in filing taxes and thanks in advance.

If you are single, then for federal you will file Form 1040NR or 1040NR-EZ with 8843.
You are part year resident of Ohio and must file Ohio tax return.
Read about F1/H1-B tax filing: Your U.S. Tax Return: The U.S. Visas