I hope your doctor knows a lot about Lyme -- that disease has wreaked havoc on my family. I've had it three times, and two of my sons apparently had it for a few years before being diagnosed. My oldest is having to repeat 10th grade b/c it so affected him cognitively (and we didn't realize it for two years). My youngest had almost no symptoms (he had a couple of hours of bad knee pain that went away) and given my track record, I went ahead and had him tested -- yup, positive.
Too much information -- but it is a tricky disease by itself, let alone adding diabetes into the mix. Our doctor uses regular medicine as well as herbs/supplements, and this approach has worked very well for us.
I was very curious, so I googled it. According to Drugs.com interaction checker: MONITOR: Tetracyclines may enhance the hypoglycemic effect of insulin by increasing tissue sensitivity to exogenous insulin.
Sounds like the doxy is the culprit. Most doctors prescribe doxy because it treats both lyme and erhlichiosis (in case you've been infected by both). If you've just been infected, amoxicillin will also work. Make sure it's a high enough dose.