To my knowledge, the NY constitution allows for someone to run for more than one office. I believe, however, that there is a law in place forbidding the holding of two or more state offices, something that is not true in NJ (where many state reps/senators are also Mayors, Attorney Generals, etc.).
In Texas, it is actually required that any state office holder seeking election to another state office or to federal office must resign the currently held state office.
This really came in handy in my county (Nueces). In 2006 the incumbent DemocRAT Sheriff decided he wanted to run for the vacant (due to retirement) office of County Judge (it's actually an executive position not judicial), so he resigned his position as sheriff less than halfway through a term that would have lasted till January 2009. Then in the summer of 2006 the US Department of Justice announced it would remove all federal prisoners housed in the Nueces County Jail due to awful conditions inside the jail. This hurt former Sheriff Larry Olivares, because it was a direct attack on his executive competence in running the jail, and it hurt the appointed incumbent sheriff who was running for election for the remainder of the 4 year term. It also hurt the county, because we lost millions of dollars in fees for holding federal prisoners and had to spend millions more to fix the problems. On election night in 2006 while Republicans nationwide were getting slaughtered, we actually elected a Republican County Judge and a Republican sheriff for the first time in history. Not only did Republicans take over control of the county, but we ousted the Robstown mafia from its control over the county. The new Republican sheriff has reformed the operation of the jail, and federal prisoners are again held there.
On a side note. The election of a Republican sheriff had the benefit of snaring a local crook named Maurico Celis who has been indicted for impersonating a law enforcement officer, practicing law without a license, and money laundering. Celis was arrested in September 2007 after he, wearing a bath robe, chased a naked woman into a convenience store near his house in Corpus Christi. He tried to get custody of the woman flashing a Duval County Deputy ID card to the clerk at the convenience store. The clerk refused and called the police. The police arrested Celis. Sheriff Kaelin called up the Duval County Sheriff and found out that Celis' Deputy credentials expired over 4 years previously in 2003. If Celis' old buddy Larry Olivares had been County Judge and a DemocRAT Sherrif elected in 2006, Celis' law enforcement credentials would never have been investigated. Celis would still be making a living referring lawsuits to a cartel of trial lawyers in Corpus Christi and still be one the top 100 contributors to candidates for state office in Texas. Interestingly, Larry Olivares is currently employed by Maurcio Celis.
Further investigations were made in October into Celis' credentials as a lawyer when a local trial lawyer in Corpus Christi started airing ads saying that Celis did not have licence to practice law in Texas looking for other plaintiffs for a class action suit against Celis. Celis then claimed to be a lawyer in Mexico, but no one has ever been able to show that he ever graduated from a law school in Mexico or ever licensed to practice in Mexico. The other partners in his law practice had to set up a new law practice without Celis.
Celis was making a living referring cases to a cartel of trial lawyers including Mikal Watts. Of course now that Celis' credentials as a lawyer are in dispute and will be proven to be fraudulent in court, the referrals are illegal, because in Texas only licensed lawyers are allowed to receive compensation for referrals of clients. As a result, of this metastasizing scandal, Mikal Watts had to abandon his candidacy, after spending $8 million of his own money, for the DemocRAT nomination for the US Senate seat held by John Cornyn. Also two local freshmen state representatives )one of whom is the son of of the US Congressman representing the 27th Congressional District) have been asked by county Republican officials to give back all the money donated by Celis to their campaigns. State Representative Juan Garcia gave back "unspent" campaign funds but did not give back $17,000 in donations from Celis plus thousands more from Mikal Watt's PAC to which Celis donated. BTW, Juan Garcia is an old Harvard Law School friend of Barak Obama and campaigned for him much of this primary season. Representative Solomon Ortiz, Jr. the son of Congressman Solomon Ortiz gote $25,000 from Celis in 2006. A third state representative, Abel Herrera has gotten $50,000 from Celis since 2004. Celis also donated $2,300 to Hillary Clinton in 2007.
As you can see the Texas law requiring state office holders to resign from office when running for another office can really shake things up by creating a domino effect. I can just imagine how corrupt things must be in New Jersey with politicians who can hold multiple offices simutaneously.