My workplace had two single stall bathrooms in the back section of the building, off the main floor, marked womens and mens, of course. We also had two sets of large multi-stall mens & womens on either side of the main floor. (Call Center, about 800 people...)
When a fellow Supervisor and I informed management of our plans to 'come out' (me as FTM and my friend as MTF), we discussed this. I was actually the one who suggested that we make BOTH of the single stall restrooms unisex - I'm a very firm believer in the concept, and some women had long been using the mens room anyway when there was a line for the womens, (myself often setting the example!

).
We also had agents who were new mothers and needed a place to breast pump - making the single rooms unisexual freed things up for them as well. Everyone - male & female - without regard is now freely using the two single rooms.
I'm the only Transperson who uses the unisex rooms exclusively, and that was by my own agreement. My fellow Supervisor had her name legally changed, and company policy says you use the bathroom that matches your legal name. Since I haven't been able to do that yet, I agreed to the idea.
In addition, I knew that while 99% of the women at work have no problem with our MTFs, that there might be some very awkward moments for the men if I started using the mens bathrooms. As I'm in management, I didn't want to expose the company to any liability since my legal name is still female. So until I get it changed (long story, don't ask...), I'm in "no-mans or womans land."

While it could be viewed as degrading and Jim Crow like, I think it largely depends on why its done, and
how its handled. In my case, it's fine, and has proved an improvement for everyone. The more work places that do unisex bathrooms, the more commonly they'll be used and accepted. Eventually, I see the unisex or "Family-room" concept becoming very common, and I think it's a
good thing personally.
Hopefully Emelye, it won't turn out to be your own 'private' bathroom; in fact, I'd be very surprised if other women at your work don't start using it frequently. I just hope that your management presents it with some positive energy and delicacy, rather than as a stigma.
Please keep us up to date, and good luck!
Scott