Asylum

I am wondering if someone knows if there are only a few specific states in the US with Asylum Courts and where those might be. Also, any info on the political asylum system would be helpful.

Well, there are no asylum courts, as such. There are regional asylum offices in San Francisco, Los Angeles, Chicago, Rosslyn, VA, Queens, NY, Houston, Texas and Lakehurst, New Jersey. But those offices cover the entire US and do interviews in a number of major cities. If you apply and fail, you might well wind up in an immigration court, but those handle all immigration cases, not just asylum. If you intend to apply for asylum, you have to either show past persecution or a well-founded fear of future persecution based on one of the five grounds for asylum recognized in international law; race, religion, political opinion (actual or impuned), nationality and membership in a social group. The last is particularly difficult to define, but exists. You can fill out the forms yourself, or obtain an attorney. There’s a fair amount of information on political asylum at this address:
http://www.uscis.gov/portal/site/uscis/menuitem.5af9bb95919f35e66f614176543f6d1a/?vgnextoid=63e26138f898d010VgnVCM10000048f3d6a1RCRD&vgnextchannel=3a82ef4c766fd010VgnVCM1000000ecd190aRCRD
And check out uscis.gov for the application form, the I-589 or go to your nearest uscis office.