wouldn't have let me get the transgender info as easily

Only ask one question at a time. If you wanted info about "transgender" then ask a "transgender" question. Example:

Are you transgender?

• No
• Yes, F->M
• Yes, M->F
• Yes, but I prefer not to specify
• Prefer not to answer

Of course, this logically excludes those who would prefer to answer but are Yes - other, but your earlier point about 'other' applies if you don't want to code open-ends.