Any reasonably competent PCP should be able to help you with this. The treatments for the 2 herpes simplex viruses are well established and highly effective, although we have no cure. The basic drugs are aciclovir and penciclovir. The first can be taken orally or used topically (or IV). The latter is only used topically. Topicals do not work very well but are available OTC - I don't know the name but I'm sure it can be googled.
They should be taken orally if you wish to control symptoms. The problem is that both are broken down rapidly and not well absorbed, so only a bit gets into your system intact. With aciclovir, you can take more of it more often to compensate. A better option is to use prodrugs (chemicals that are converted into the relevant drug in the body, usually by the liver). In this case, the prodrugs are stable and easily absorbed at high concentrations.
The prodrug for aciclovir is valaciclovir and the prodrug for penciclovir is famciclovir. These are the best drugs to control recurrent outbreaks of the herpes simplex viruses. They can be taken either to suppress (i.e. every day, whether or not you have symptoms) or to treat an outbreak (higher doses, only when you already have an outbreak). Suppressive therapy also reduces the risk of transmission to others, although it does not eliminate it.
They are generally considered to be very safe drugs. I don't have either herpes simplex virus, but I did take very high dose valtrex (1g q6h) for ME and I did not have any side effects, although at those levels it's a potential concern - but they were trying to treat HHV6 at the time (which unlike HSV1/2 is not very susceptible to valtrex). This was many years ago, and I believe that there is little merit in trying antivirals in ME at this time. It didn't really help me (although it knocked down titers on my lab work for HHV6). However if you have herpes simplex virus 1 or 2 and are getting outbreaks, regular doses (much less than what I was taking) are generally very safe and your regular doc should be willing to prescribe it.
Also - are you sure you have cold sores? Do you know the difference between cold sores and canker sores? Where are they located, what do they look like? Canker sores are found generally inside the mouth, usually on the inside lips and cheeks as well as the tongue and even soft palate. Cold sores are found usually on the lips, the vermillion border between the lips and regular skin, and sometimes on the hard palate inside the mouth or on the attached gums. Canker sores are usually white with a red halo. Cold sores start as blisters and rupture, coalescing into larger cold sores that scab over.
Canker sores are not viral and are not contagious, and they cannot be treated by antivirals. The true cause is unknown but probably immune related. Topical steroids can help canker sores but make cold sores worse. Cold sores are caused by herpes simplex 1 usually (and occasionally herpes simplex 2, but this is rare). They are viral and contagious, even when not visible/active (although less so when not visible). They respond very well to valtrex (valaciclovir) and famvir (famciclovir). The former, in particular, reduces rates of transmission to others.
If you are not troubled by your cold sores themselves, and not worried about transmitting the virus, but are looking only for a clue as to ME treatment, I'd advise you to look elsewhere. I tried it in my much younger, less skeptical days on a doctor's advice, and while I suffered no harm, I also did not benefit. High doses can be dangerous, but normal/low doses as indicated by the FDA prescribing information are unlikely to cause problems.
If you still aren't sure if you have cold sores or canker sores, ask your doctor to swab a sore and PCR it. Also, get a type specific IgG herpes blood test. IgM testing is not useful. Western blot is the current gold standard for blood testing and is both more sensitive and more specific, but not as widely available.
I'd urge you to view theories that suggest ME is a result of herpes virus infections with a high degree of skepticism. The only thing high dose valtrex did for me was lighten my wallet (about $7500 over 6 months). Fortunately it's generic now and much cheaper, and you wouldn't be on crazy high doses for herpes simplex anyways, so it's probably not a lot of money now for that indication.