Yes, there is a good chance this is a false positive.
First, the IgM is not a reliable test. It doesn't do a good job of differentiating between a new and established infection, and doesn't do well differentiating between type 1 and 2. It's supposed to look for new antibodies, but can be reactive when someone has a well-established infection. It can also react to antibodies for other herpes viruses, like the chicken pox.
It shouldn't be done on adults. The CDC has said this for a long time, but doctors aren't aware of it.
Also, if you've had cold sores before, you have an established infection. You shouldn't have the IgM test, which is supposed to look for new infections - within about 2 weeks of exposure. (Though are you sure? Cold sores occur on the lips, not in the mouth. Sores in the mouth are canker sores, not cold sores. You should google them both to look for images to see which one you had.)
In any case, the test you want is an IgG type specific IgG blood test.
When was your last possible exposure? What was your exposure?
It's hard to say if you have it or not - the IgM is unreliable enough that you should act like you didn't have any testing - ignore it completely.
Herpes is infectious, obviously, but a lot goes into how infectious it is. It's more infectious with a sore present than if one isn't. If someone takes daily meds to suppress the virus, that lowers transmission rates.
If you have genital herpes type 1, which you can get from oral sex, that's less infectious than genital herpes type 2, or oral herpes type 1.
So let's cross that bridge when we get there. Get an IgG type specific test. It can take up to 4 months to be conclusive, though most will show positive by 6 weeks.