All you really need to know is how long you lived in each state, and how much tax you paid to each state (from your W-2) to file your Part-Year Resident Returns for New Jersey and New York. Prepare New York first, then New Jersey.
For example, New York will look at your total income, then figure the tax on that amount. Then they will take 25% of that tax (for 3 months you lived there) and that's how much tax you owe them. Then they will look at the tax you paid them (as shown on your W-2) and figure out if you owe or get a refund. TurboTax will do the calculations for you.
For New Jersey, you report the income you earned while living there. It sounds like your W-2 has the correct amount (about 110K; 150K/12 * 9). They calculate tax on that amount, then figure if you owe or get a refund (based on NJ tax paid on your W-2). You really shouldn't need to do any calculations with a W-2; TurboTax will do them for you, as long as you reported the dates you lived in each state.
Here's some info on Allocating Income as a Part-Year Resident.
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