This situation is difficult, but not impossible. There may be many solutions available, but more information is needed. It is extremely important that you meet with an immigration attorney who has experience with complex cases such as this one. If you already have a criminal attorney, that attorney and the immigration attorney can work together to find the best plea for the criminal case that will have the least amount of impact on the immigration case.
You need to hire an attorney right away....
Unfortunately a false claim to citizenship is an unwaivable bar to adjustment of status. Perhaps you can get a copy of the I-9's to be sure that your partner did, in fact, claim to be a US citizenship instead of saying that, for example, he was a lawful permanent. Regardless you should absolute utilize the services of an attorney before making any decisions.
If you are now placed into deportation (removal) proceedings, then there are some defenses for which you could apply that allow you to apply for a work permit. Now that you have had contact with immigration it is IMPERATIVE that you speak with an immigration attorney who has extensive experience with defense against deportation.
There are a lot of things going on here that make it imperative that you speak with a lawyer. Right now, more than ever, you MAY have an opportunity to get your husband his papers quickly due to an almost unprecedented occurrence of visa availability. You should schedule an appointment ASAP.
It is VERY important that you know your rights, especially regarding your right to remain silent and your right to not testify to anything that could incriminate you in court. Though it is irrelevant to the hearing, a judge COULD ask about your immigration status and you should be prepared how to answer.
Also, it is important to know that immigration (ICE) does come to the court to talk to people as they leave AND they sweep the jails every day. So please, speak with an attorney to prepare you...
If your student visa is still valid and has no limitations regarding travel, that should be all that you need to leave and then reentry the country. It is important to speak with an experienced immigration attorney before leaving the country.
Whenever there are criminal issues involved, it is necessary to speak with a lawyer. Many crimes are considered misdemeanors by state courts but are considered felonies by immigration. The intersection of immigration and criminal law is very complex. You should speak with any attorney before making any decisions whatsoever.
It is important that you meet with an attorney dedicated to immigration law to discuss this in more detail. The answer here would depend on many factors, but the important of the factors is whether the first marriage was deemed to be fraudulent, which could prevent any further petitions. However, that is not ALWAYS the case, so you must speak with an immigration attorney to make a plan of how to go forward.