If you're a full year resident of New York, you're going to report the $10,000 capital gain on your New York return and then apply your New York capital loss carryover to that gain, if any. If your New York capital loss carryover is the same as your federal capital loss carryover, you'd be able to utilize $13,000 and deduct $3,000 on your return.
If you weren't a resident of New Jersey, you wouldn't need to include any of the gain on your New Jersey return unless the assets you sold were New Jersey assets. If they were, you'd only report your New Jersey sales and New Jersey capital loss carryover to New Jersey.
