You most definitely need to get help for your anxiety. There are a number of things you can do to get it under control. All of these have worked for me:
1. Change your diet. Cut out processed foods and starchy carbs (potatoes, rice, pasta, cereal, most breads). You'll be able to concentrate a LOT better and will have less anxiety.
2. Get on a medication that can help with anxiety, but not make you foggy. Anti-depressants are no good. When my anxiety was pretty bad, I took propanolol, a beta blocker (it helps regulate your heartbeat but has the added bonus of controlling anxiety). I took it for a while when needed and felt much calmer.
3. Try 5-HTP. It is a natural building block for serotonin, which is key for anxiety (not to mention sleep, pain, depression, etc.). I recently started taking it for chronic pain, but it has mellowed me out considerably. I sleep harder, so I have to make sure I take it at least 2 hours before sleep, but it doesn't make me drowsy or foggy during the day. I would recommend researching it.
All of that said...If your anxiety is so bad that you start crying during the LSAT, then you NEED to get on medication. Therapy might help as well, but anxiety is less an emotional thing than a brain thing, and there are medications that work without making you feel like you're floating. Go see your doctor again. If you give yourself time to get used to the medication before you take the LSAT, you'll be ok.