Does your W-2 show New Jersey wages in Box 16? If so, your company probably split your income the right way.
New Jersey residents are taxed on income they earned while living in the state. Your relocation and signing bonus, if received as a lump sum, should all go to New Jersey.
July should be prorated based on where you were living when you got paid. You can refer to the year-to-date earning on your pay stubs, although those may include non-taxable income.
Washington state has no income tax so you will only have to file a part-year New Jersey tax return.
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