Sounds like A LOT of people may have Lyme! Honestly, I don't think 2 to 4 weeks is that long to wait.... given my experience. I can't see my new rheumatologist for fybromyalgia until the end of August and I made the appointment a week ago.
Hope this helps.
You could make an appointment with several doctors and ask to be put on each one's cancellations list. Then you can take the earliest appointment. If it goes well you can cancel the others. If it does not go well, you have a back up. Just be sure to cancel any appointment you do not go to or you may be charged for them.
Maybe you could ask your doctor to extend your Doxy while you are waiting to see a specialist.