Can you remember the name of the lip balm you used? Do you have any allergies? Have you investigated whether you have any deficiences or had any blood tests? Maybe if you get the brown spot removed, you can request the doctors do a full biopsy of possible pathogens.
And by "leaving them alone," have you let the skin build up and not removed it, or have you just not used lip balm for 10 months?
Regardless, your lips look pretty good. I would avoid the balm and petroleum and just don't forcibly peel the skin until it comes off easily. That's just my advice.