If I had had a unisex birth name, I think I would've been very conflicted about it and maybe would've only changed my middle name. My concern would definitely be that people familiar with me would have a harder time mentally making that switch to male if there isn't a new name prompting them to do so.
So if it were me, I think it would depend entirely on environment. Are you still living where you grew up, or with family/friends? Are they supportive of your transition and already using male pronouns? If you haven't come out to them yet, do you think they would be supportive and use the right pronouns?
My whole plan was and is to make a big geographic move after I'm 100% done medically transitioning. And if my birth name had been unisex, I wouldn't have worried so much about changing it because I would eventually be surrounded by people who had no idea that people used to use female pronouns with me. If that makes sense.