It certainly doesn't sound like a cold sore. A cold sore (herpes, fever blister) produces a cluster of blisters which scab, last 7-10 days, and go away. They may recur later in more or less the same place and do the same thing.
What they don't do is 1) "never completely [go] away", or 2) get "a little bit of a recurrence," or 3) "spread just a little bit to other areas above my lips."
I think you may have just had some mild irritation, which you are concerned about and therefore keep rubbing at. Although you say, "I never touch it," you also say, "Some days I wake up and I can peel tiny bits of skin off."
Please stop peeling tiny bits of skin off. Instead, apply over-the-counter 1% hydrocortisone twice a day and leave the area alone.
If that doesn't make the redness disappear in 3 weeks, I suggest you see another doctor, who is perhaps more familiar with herpes and cold sores. I say that because your PCP wasn't sure, as you say, and ordered blood tests, which are not very relevant. Finding herpes antibodies in the blood doesn't mean anything more than that you were exposed sometime in your life to the virus. Finding no antibodies means that it couldn't be herpes. So if "Blood tests were taken and I seemed to be fine," that should have ruled the condition out, not in.
Best.
Dr. Rockoff