You are correct. Since you are working for your own convenience, you do need to file a NY return claiming the income you earned while working remotely for NY.
Your company likely withheld based on your address. You do technically have a tax filing requirement for NJ too so that is why they likely withheld for NJ instead of NY. You will want to contact your employer so they withhold for both states.
To avoid paying double taxes, you will complete your NY return first and then you will take a credit for taxes pain to NY on your NJ return. You will be asked this as you walk through the NJ return. The credit you get may or may not result in a refund, it simply depends on if your taxes to NY are more or less than they are to NJ.
NJ will not give you a credit of more than your tax liability on that income to NJ even if NY charged you more.
You can use the tax liability to NY and NJ to ask your employer to withhold accordingly. You will need to fill out a new NJ-W4 and an IT 2104 This will help you avoid the penalty for next year.
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