New Jerseyans working across state lines pay taxes on their income earned in New York to New York. New Jersey also taxes that income but gives residents a tax credit for taxes paid to other jurisdictions.
Complete your NY non resident return first. TurboTax will do all the calculations for you.
I'm not sure what credit you are asking about, or why you think you might not "still" get it. On your New Jersey resident income tax return you get a credit for part of the income tax that you pay to New York. In TurboTax, prepare your New York nonresident tax return first, then your New Jersey resident tax return, to make sure that the credit is calculated correctly. The credit is called "credit for income taxes paid to other jurisdictions." There is no change in this credit for 2021. If you got it in past years you will get it for 2021.
NJ follows the general rule.
The general rule is: your report all your income on your home state return, even the income earned out of state. You file a non-resident state return for the state you worked in and pay tax to that state. Your home state will give you a credit, or partial credit, for what you paid the non-resident state. You will have to file a non resident NY state return and pay NY tax on the income earned there.. You will also file a NJ full year resident return and calculate tax on ALL your income. NJ will give you a credit, or partial credit, for the tax you pay NY. So, there will be little or no double taxation, but you have the cost and hassle of filing two state returns. Do the nonresident state return first.
If you work in a state without an income tax (e.g. Texas or Florida), there will be no credit, since there was no FL or TX tax. In other words, having worked in a state without an income tax does not get you out of paying state tax on that income, to your home state.