From your post it's not clear whether it's you or your spouse who worked remotely for the New York organization, or was it both of you? You wrote:
"My spouse worked remotely"
"i would work remotely"
Assuming it's you, if you never go to the New York office of the organization, you do not have to pay New York tax on the income from New York. Since Texas has no state income tax, you would therefore pay no state income tax on the income you earn for working remotely in Texas for a New York organization. However, if you go to the organization's New York location for even one day, then all of the income you earn for the remote work is considered New York source income. You would have to file a New York nonresident tax return and pay New York tax on all of the income for the remote work.
