Yes, you do need to fill out a nonresident tax return for New York. Even if you never set foot in NY, you are still required to pay NY income tax on income earned from an employer there. It's called a "convenience of the employer" rule and NY is one of the states that requires its employers to withhold and the employees to pay NY income tax even if the employees are based elsewhere. Annoyingly, you are required to file in NY even though you only earned $270 as long as your federal gross income exceeds the NY standard deduction for your filing status.
Texas does not have a rule like this, and it would not matter if they did because they don't have an income tax. You are not required to file anything for Texas state taxes.
The freelance work is considered self-employment and would have to be reported on your return as such. On the plus side, this means that any ordinary and necessary business expenses you incurred as part of this freelance job would be deductible as a business expense. The downside is you have to pay self-employment taxes on this income in addition to the state and federal income tax you normally pay.
As your resident state, NJ taxes all your income. The $270 from NY, the 1099 from TX, all of it. Whatever tax you pay in NY will be credited to your NJ taxes so you will not double-pay tax on the $270 of income even though both states tax it.
