The issue is that doing this in the order you would like to do it in is going to be very difficult, bordering on impossible. What you've said is that you need to (1) move out of your parents' home, (2) get an autism diagnosis, (3) utilize social services programs to find employment and housing, and (4) transition in that order. Correct me if I have that wrong and I'll make a new post with updated thoughts.
With these goals in that order, it does not seem achievable based on the information available. Your ability to do #1 (move out) hinges on your ability to do #3 (access social services programs for employment/housing), which you can't access until you've achieved #2 (gotten a diagnosis). Unless you have a steady income stream, substantial savings, or people who can house you, it seems like you're going to have to re-do that order if you're going to start moving forward.
Be explicit about what's going on with your parents or what you'd expect to go on with your parents. From what you've posted so far it sounds like if you were to move forward with getting a diagnosis behind their back, they might: be annoyed that you used their insurance for it, yell at you about it, and talk down to you about it. And if that is the case, then my response is - so what? Let them be annoyed. Let them yell. Let them talk down to you. If the end result is that those things are all you'd have to suffer through in order to get a diagnosis that would enable you to move forward with the rest of your plans and in turn the rest of your life, so what? You can get over words and negative feelings, especially if it means that in doing so you'd be setting yourself up for greater success long term.
If the anticipated response from them is something more than that (you think they will become physically violent, you think they will withhold food from you, etc.), then you need to speak to someone in social services in your county because you're living in an abusive situation and there are already services available to help you without an autism diagnosis.