I agree with you this does not sound like a cold sore, which would be more circumscribed and rounded and would come and go. What this sounds like instead is a condition called angular cheilitis or angular stomatitis, which is an inflammation and persistent cracking of the corners of the mouth.
Most cases of angular cheilitis are easily treated, because it turns out they are closely associated with infection in the mouth with thrush, a common yeast organism! After your doctor
diagnoses the condition, they will likely want to try treating it with a topical anti fungal cream, which is usually very effective. Occasionally, the angular cheilitis is caused by a bacteria instead of a fungus, in which case a topical antibiotic cream rather than an anti fungal cream is used.
Some cases of angular cheilitis begin because of a nutritional deficiency, especially B vitamin deficiencies and iron deficiency. For these reasons, it is always worth having a thorough physical examination from your primary care doctor
when dealing with this condition before simply brushing it off as a thrush infection (some cases may in fact be infected with thrush but may have begun because of the nutritional deficiency). Talk to your primary care doctor about this!