I don't know if you're still looking for information about the Emirate, but I'll just go ahead anyway. (I found this on Google.)
First of all I am very reserved on calling us Persian, as an Abbasid is exclusively Arabic and although we were greatly influenced by Persian culture, we identified as Arabs and practiced strict Arab customs and traditions.
Anyway, back to the case at hand.
After Hulagu sacked Baghdad after the treachery of Ibn al Alqami, Abbasids dispersed across the Middle East. Some returned to their ancestral homeland of Hedjaz, others fled to the Mamlukes in Egypt, and a distinct number of Abbasids (probably around 1000-1500) fled to Southern Iran where it was relatively safe from the Mongols who were slaughtering Abbasid princes left right and center.
Initially, the Abbasids settled in the Sunni Persian city of Khonj and were revered because of their Hashemite ancestry and theological knowledge. They were often assigned as judges to solve domestic disputes.
The Abbasids eventually moved to Bastak where they made it as a center of rule over the willing Sunni Persians.
It was a relatively prosperous and peaceful reign of rule and they had good relations with the Shah of Iran. Little by little, however, the Safavids grew larger and larger and although Abbasids had some support from the Ottomans, it could not overcome the new strong Safavid state.
As a result, my ancestors came back to the Arabic Gulf countries with many other returning Arabic tribes. Currently the descendants of the Bastaki Abbasids reside in Bahrain, Saudi Arabia (me), United Arab Emirates and Kuwait. Probably for the best too because of how racially Arabs are treated in Iran.
That was a general outline of things. There are many Persian and Arabic books that describe this Emirate, however, there are none that are in English.
Sorry for the poor sentence structuring, I'm still trying to get a grasp on writing in English.