Use your married name and include maiden name where it asks for other names used. You are fine to submit the A/S application.
The answer provided is general in nature and should not be construed as legal advice as not all facts are know by the attorney, nor does the answering of this question create an attorney client relationship.
You can use either your maiden name or your married name. If you use your maiden name, be prepared that all the paperwork, including the green card, will be issued in your maiden name. If you want to use your married name, you can do so as well but make sure to put your maiden name under other names/aliases category. If you have time, go to the Social Security Administration and change your SSN number as well as her Driver's License into your married name. This is not a pre-requisite but it will ensure that the USCIS is not going to delay processing because the records do not match.
Also, since you got married within the 90 days, your status is fine- just file for adjustment without delay.
one of the few black line rules in immigration is that you must wear your spouse within 90 days after admission on the K-1 visa. There is no deadline for filing for your adjustment of status based on that marriage.
Regarding your name, when you fill out the immigration forms you will tell them your name. Choose the name you want carefully. You may either keep the name you have, or you may adopt for spouses last name as your own. As this is true for hetero couples this should be true for homosexual couples as well.
I am a pretty smart guy, but I do not know everything. My answers here come from the knowledge I carry around in my head--if you want me to look something up, you should come see me. Take any advice here as being general in nature. Please be careful with your life and make decisions that are smart.