Welcome, and sorry you had to come here, but you've come to the right place. It's good your neurologist has you on doxy, but six weeks of doxy is not going to get you well. The fact that you had a bulls eye rash and markers for lyme is proof enough that you have lyme disease. Do you recall being bitten by a tick back when you got the rash? If you've had it for 13 years you'll need much more aggressive treatment than what you're getting now.
The very first thing you need to do is set up an appt with a Lyme Literate Medical Doctor (LLMD) and get on the appropriate protocol for treating your disease. You are right, lyme could very well be passed on to your son. You are wise to have him checked as well. I strongly suggest you have him tested by a LLMD.
If you will click my email icon and send me an email I will be glad to give you some links for locating LLMD's in your area. Let me know what state you live in and I might be able to give you specific names of LLMD's in your area. If not, I'll provide you with links where you can get names of LLMDs near where you live or anywhere in the US for that matter.
You've possibly have had this disease for a long time, it's so very important you get the appropriate treatment and the best way to do that is to see an LLMD as soon as you possibly can. Also, if your son has it (I pray not) the sooner you get him treated the better his chances are for a quick recovery.
Please email me for more information if you wish. You are in my prayers.