It is very likely that you will be able to obtain your LPR status through the CAA. It used to be that the spouse of a Cuban LPR could apply for adjustment of status under the CAA if the spouse met all the requirements of the CAA regardless of how the Cuban spouse obtained LPR status, however, recent BIA decisions have restricted eligibility to spouses of Cuban LPR's who are CU6 adjustees. It sounds like in your case your Cuban spouse will also adjust status through the CAA so you should be OK.
Whether you can request your green card through the Cuban Adjustment Act will depend on how you entered the United States. When you say you entered illegally from Cuba, do you mean you left Cuba without documents to travel or that you entered the U.S. without inspection? In order to get your green card through the Cuban Adjustment Act, you must have been inspected and admitted or paroled. If you have an I-94 or other proof of lawful entry, you can most likely apply for your green card under the Cuban Adjustment Act. If you entered without inspection, you will have to wait until your spouse becomes a Lawful Permanent Resident or U.S. citizen. He/she will have to petition for you through the standard family petition (I-130) and you may not be eligible to adjust status in the U.S., but rather may have to consular process due to the unlawful entry.
The information offered is general in nature and not meant to be relied upon as legal advice. No client-attorney relationship is created through this information. Please consult an attorney prior to making legal decisions.