Just want to agree with Ema's advice, to get re-tested before taking this too seriously. To have an IgM only positive is a bit strange for a chronic CFS case unless you recently were infected.
Also, if you do pursue doxy, I suggest following the Garth Nicolson protocol for multiple chronic co-infections, which includes regular breaks, probiotics, etc. I'm speaking from hard experience as I took doxy for a full year and developed a host of problems as a result, I did not follow Nicolson's protocol but wished I had. The problems I developed included a swollen pancreas (known side-effect with long-term use), and severe digestive dysbiosis that led to a worsening of my CFS. However, for the first 6 months or so life was wonderful, I thought I had found the cure.
The problem with treating Lyme as the cause of CFS is that by the time you have CFS, the Lyme is usually disseminated and nearly untreatable. 30 days of abx won't touch the problem, those spirochetes are in tissues and even bone where the abx (delivered by blood) can not touch them. Also, they are cycling through multiple forms, and are sometimes not vulnerable. In my own case, with a positive test for Lyme 10 years ago, I eventually settled for a disease management approach. I try to keep the Lyme dormant, but mostly using natural abx.
FWIW, minocycline did nothing for my own Lyme infection, only doxy helped, but I did not want to stay on that. I did try a short course a few years later, after my pancreas was working better again, but it did not help like the first year.
Some natural abx that helps me: GSE, oregano oil, olive leaf extract, colloidal silver (I just use that occasionally), and chlorine dioxide solution (CDS, a new form of MMS). The CDS is probably the best of all of them, but it also only helps to a point. Another natural treatment has been Kyolic Immune Formula, which boosts the NK function with aged garlic and mushroom extracts, and actually gives me more energy. Unfortunately, RIFE does not seem to work for me, except it aggravates symptoms for awhile.
I know your doctor is rushing you but if you've been sick for decades, and Lyme is part of your pathology, you are probably way past the short time period where a quick course of abx could solve this.