Frankly, I am not at all sure you have committed absolute fraud or done something wrong here. Because you could not have filed for the week until after it passed. Did you call every day during the week you were out of town, or had they told you they probably wouldn't have any more work for the rest of the week the last time you called before you left on vacation?
Since your union is your sole source employer, it would not be unreasonable to have you check for work one time at the beginning of each week. Nobody would require that you check with them every day. That would be somewhat of an overly demanding job search requirement. Of course you might want to check with them every day, that's your prerogative.
If you were told there was not going to be any more work available for you until say, the following Monday, then you pretty much free to leave town or do whatever you want to, as long as they, your sole source employer, does not contact you and offer you work which you refuse.
Many companies sign their employees up for unemployment while they are closed down for Christmas or summer holidays. These employees are "job attached" and do not have to register for other work or make a job search, other than being available for work if it were to come available for them from their permanent employer. They are not obligated to stay in town.
You say you did not receive any money. Your first week was your waiting week, on a new claim, right? So why did you not get paid for the second week? It sounds to me as if you should have been. You need to call and talk to the system anyhow. If as you say, you SOMETIMES get to work, you should be receiving unemployment benefits for any week they do not have work available for you. Or any week that you only work one or two days and do not receive as much gross wages as you would have made in weekly U.I. benefits
As I said, call and discuss it with them generally without saying, "I just found out I was supposed to ...blah blah blah. " By the way, "I just recently found out" you say that you did this wrong. Remember, no one can tell you for sure the answers to this question or what you are actually supposed to do except someone who works with the NY unemployment system. Your friends, co-workers, etc, are not reliable sources of information about claims.