My ex-husband has offered me a GET, however he is ignoring a judges decision underpaying my alimony, and so I have not accepted the GET yet. I am willing to do so once he lives up to his responsibilities. If my ex-husband manages to dissolve our Jewish Marriage prior to my accepting of the GET, where does that leave me?

Your question describes an interesting situation-- It appears that you have a civil divorce but not a "religious" divorce. There are two parts to this response:

A. First to answer your direct question-- It depends on what mechanism within Jewish law is used. If your ex-husband dissolves your Jewish marriage without your cooperation in a recognized Orthodox Rabbinical Court in accordance with Orthodox halakha (Jewish law), then your marriage will be dissolved and both your personal statuses will then be single and free to remarry. However, that would only be the case if your original ceremony was somehow annuled. There is another possibility within Jewish law that would allow him to take on a second wife without you having accepted a "get". That is called heter meah rabanim or heter nissuin--where a Rabbinic Court may permit him to remarry and that marriage would be valid both civilly (since he is divorced) and religiously. When this is done in the Israeli Rabbinic Courts the heter will be issued only after the husband deposits a "get" in the Beth Din to be held in escrow for the woman to collect whenever she decides to do so. In the US no Rabbinical Court should ever issue a heter without similarly arranging first for holding a "get" in escrow, but there are indeed Rabbinical Courts who do not do so. This presents a danger for you due to the fact that when a man avails himself of that solution to his wife's non-acceptance of the "get", he will have no incentive to ever give her a "get" and she will remain his wife according to Jewish law -- thus unable to remarry herself. In other words, by refusing to accept a "get" now you may eventually turn yourself into an agunah.

B. As part of the civil proceedings your husband has not lived up to his financial obligations. Both theoretically and practically this grievance should be addressed within the legal framework of the civil system which is the source of the obligation. Even though you believe that not accepting a GET is justified, this decision is a "mixing-up" of two legal systems: civil as opposed to Jewish law. (For example--if you are stepping into the field of Jewish law you should be aware that there is no such concept of "alimony" for a wife after a marriage is over.) As a matter of ethics and sound practical policy, the "get" should never be used a a tool for leverage in either direction.

Thank you Rabbi Levmore.
My ex-husband has already gone thru with his portion of the GET process.  Are you saying that If i do not accept the GET now, 
I may lose my ability to obtain it in the future?

I cannot answer your question accurately since I do not have all of the information. For example:

1. which steps exactly has your ex-husband taken?
2. In which Rabbinical court? Is it Orhtodox? Is it a recognized Rabbinical court? If you are not sure of the latter, you can call the Beth Din of America and ask them if this particular Court is an Othodox recognized one.
3. If indeed your ex-husband has deposited an actual "get" in the court to be held in escrow for you, what is the court's policy? Do they summons you to come and get it? For how long will they hold it for you? indefinitely?
4. Did your ex-husband attach any conditions to the giving of the Get to you by the Rabbinical court? Can he renege on it and thus invalidate it ,according to the Court's policy before it is delivered to your hands??
5. If your ex-husband did not actually deposit a GET in escrow for you--then he may at any time decide that he is not going to give you a Get at all.

For questions 1, 3 & 4 you may have to phone the Court directly to receive the information.
I suggest that you try to find out the answers to questions 1 & 2 and then call the Beth Din of America. Under certain conditions you may be able to get them involved as a Beth Din that you would want ot use. They may be able to negotiate the alimony issue if you would be interested in that and your ex-husband would agree.

Don't be afraid to ask questions of any Beth Din/Rabbinical Court.

The only way to find out which rabbinical court is to ask the rabbi who is holding the get for me?
Is there any other way to get this info?

How do you know that your ex has begun the GET process? Has he told you about it? Has a particular rabbi called you? Did he leave his name and number? If so, have you googled him? What is his name and location? where do you live? Where does your ex live?

If a rabbi is "holding the get for" you then he must be part of a Rabbinical Court. If you know his name, then call the Beth Din of America
 and ask if this is an acceptable rabbi/rabbinical court. In other words--does the BDA honor "GET"s arranged by this rabbi. The BDA is the most Central authoritative Rabbinical court in the US, whose GETs are fully accepted by the Israeli Rabbinic Courts.