I'm not sure how to file these numbers. When I enter the numbers as turbo tax requests, it says I owe an additional 4% in taxes to the first state
Payroll with multiple states in a known issue with the IRS.
The W2 is showing the full income earned where you originally lived and worked plus the amount earned while living in the second state.
You will have to enter the correct amounts into the program on the W2 state line. How much was actually earned in the first state? If it was NJ or CA, I would contact HR and ask them to determine the amount earned in that state.
If you are lucky, you can just subtract and the difference will be the first state's wages.
Here are the steps you should take: