Check your state and county requirements. For example, in CA, birth certificates are not required in all counties.
In my county, it was strongly recommended, so I brought it along, but I wish I hadn't. This was pre-transition, but I had already changed my name by the common usage route, and that meant that I had to bring my name change document to explain why the names on my BC and CDL didn't match. Since I hadn't changed my name through the courts, I had to do a powerful lot of persuading that my name change was indeed legal.
Your situation is obviously a little different, and I'm not sure of the ramifications. If you have M on your state ID but F with the Social Security Administration, you can probably get married in your state if you can find a county that does not require a BC, but I believe that you won't be eligible for the usual federal benefits that go along with a traditional marriage. You might not even be eligible for state benefits, such as the right to file taxes together...I guess it all depends.
In most cases, a passport is as good as a BC. Do you have M on your passport?