fill the name that you would like to show on your greencard and you would like to use in your future on every forms that you are going to submit to adjust your status to permanent status ja. Once you obtain greencard, you can bring it to change your name on your marriage certificate, your SSN, your driver license, the banks, or etc ja. If you use your current full name on every forms when applying to adjust status to permanent resident, then your greencard and work permit will be your current full name na. If you want to change it to the new name later, you would have to submit the form to USCIS to change your name which will take a few months and then you would have to correct your name to the new name later anyway na ja