I feel your pain, both for the headaches and not wanting to disrobe for a massage.
If you are receiving a neck/shoulder massage, you will remain fully-clothed, and sit in a chair specifically designed for that type of massage. You will be leaning forward with your head in a "donut-shaped" head rest. The massage therapist will work on your neck, shoulders, and upper back, and possibly your scalp.
If you are going for a full-body massage, you will be asked to lay down on a table (possibly both face-up and face-down, and different points in the massage) and cover up with a sheet. In this instance, because the therapist requires access to more of your body, you may be asked to remove your shirt and any undergarments.
HOWEVER, the massage therapist is being paid to help you, and if you aren't comfortable taking off too much, tell the therapist. If you don't want to take your shirt off at all, wear something thin, but be aware that the massage may not be as effective. Another option would be to wear a tank top under your shirt, or a sports bra, and take your shirt off while showing as little as possible.
I saw a massage therapist for about 6 months, 2 times a month, give or take. It helped immensely with my migraines. It can (and does) work, so try to remain open-minded about what may be necessary to get the relief you're wanting.