Zak: There are a couple of ways to go about this.
1) You can come out at work and start going by your new name now, regardless of whether or not you’ve had a legal name change. This can be tough because you can potentially run into issues with on the job discrimination and harassment, even being fired, and even if you don’t it can be quite awkward. Still, this would allow the people at your job to get used to you as B and increase the chances that they will use the correct name if contacted as references. When asking to go by B you should stress that even though it is not your legal name you plan on eventually changing it legally and that it is not a mere nickname but rather an important expression of your gender identity. How this goes will depend on your boss and your work environment, but there have been people who have done this and had it go very successfully for them.
2) You can wait until you have left your job and changed your name legally to contact your former employer(s) and let them know about your name change. When doing this you might want to say something like “I’m about to apply for a new job and I wanted to contact my references to update them on a major change in my life, I’ve legally changed my name to B and use male pronouns. This is something I’d like my references to be aware of so that they can use the correct name and pronouns when contacted.” This can be awkward and how it goes totally depends on your former employer(s). Adrian has done something similar to this and had a person refuse to use his male name and pronouns because it would be “misrepresentation” since the person didn’t know him that way at the time (he also had people who were totally understanding and cool with it). That’s always a risk, but usually if you contact someone you get a good idea how they will respond and can leave them off of your references or write “do not contact” if you know they will out you or say negative things about you because of your trans* status (this of course can create additional problems because references are extremely important and it looks bad if future employers can’t contact important former employers, but there are situations where this would be much less of a big deal, for instance if you don’t allow them to contact your boss from an older job in an unrelated field).
3) Apply for your next job as openly transgender (by saying something like, “my legal name is B but it was formerly A, as you might see if you contact my references”), or as A and then come out and ask them to call you B, or apply as B and hope that they won’t contact your references (or specifically tell them not to contact certain former employers). Basically you put yourself through all of the potential awkwardness and discrimination of the first option, but at a different workplace. If you’re open from the beginning as trans*, or if you’re applying as A with the intention of eventually working there openly as B, you can specifically pick a workplace that is affirming or LGBTQ friendly. This option is difficult, particularly in our current job climate, but it’s something to keep in mind if you think your current employer is going to make it impossible for you to move forward and not have to be out at your next job. Again, you can always apply as B with no references from your former employer, but this can also present difficulties. The best option for you will totally depend on your situation.
Basically any way you slice it this stuff is difficult and depends on a lot of different factors. As far as I’m aware there aren’t any hard and fast rules when it comes to being trans* in the workplace or transitioning, or even changing your name, on the job. Every workplace, boss, and group of coworkers are different and the non-discrimination policies and laws for different workplaces and states also vary immensely. There are likely other good ways of dealing with this that I didn’t think of that maybe someone will comment on, so be on the lookout for that too. Good luck!