If I'm under the DACA law and I'm married to a US citizen, can he fix my situation?

No, under the current immigration laws, you are not eligible to get a green card even though you may be eligible for DACA.

Because you entered the U.S. without a visa, you cannot receive a green card just on the basis of your husband's immigrant petition. In your case, the process is more complicated and involves obtaining a waiver of your unlawful entry and unlawful presence in the U.S. Your immigration attorney would have to carefully weigh many details of your personal history and family situation before he/she would be able to tell you whether your chances of success are good enough to start the process now - or wait until the Congress would change the immigration law and see if you would benefit from the reformed law.

If you have no criminal record or are otherwise inadmissible, you should be able to get your green card through your husband. I suggest that you contact an experienced immigration attorney for a face-to-face consultation and give him/her all of the facts surrounding your situation. (S)he would then be in a better position to analyze your case, advise you of your options and, if possible, make sure that the necessary paperwork and documentation is presented so as not to delay the process.

Yes, there are two ways you can get your residence through your US citizen husband: (1) you can apply based on your entry with your cousin's papers, and (2) you can travel on an advance parole (which you can get since you have DACA) and after you travel outside the US and re-enter, you can file through your US citizen husband.

I have been successful in getting an advance parole (permission to travel outside the US), then filing for adjustment upon return to the United States.