Sorry for the delay.
If I have got this right, you had two case. One is in New Hampshire, and it was a conditional discharge. You believe it may have been closed when you move to Texas.
Then you got a new case in Texas, where you are on supervision and want to know if the case would transfer.
First of all, you need to find our for sure if your case did close in New Hampshire, because in my experience when you leave the state permanently before your case is officially closed, what generally happens is that the court will issue a warrant for your arrest. If they have done that, you will not be safe from arrest in New Hampshire until you go back to court and deal with that warrant.
The way to do that would be to call the clerk of the court where your case was pending and ask him if your case is closed or if there is a warrant. He can look it up and tell you. If there's a warrant, it won't ever expire. It will stay out there against you until you or a New Hampshire lawyer on your behalf appears in court and stands before the judge ready to deal with your first case.
As for transferring your Texas supervision to New Hampshire, you would not want to do that if there's a warrant in New Hampshire. But do you have to report to a probation officer? Have you been told that you cannot leave the state without permission from your PO and the court?