Welcome to JustAnswer, and thank you for putting your trust in me!
I agree that most pain originating from a chronic dental
infection would have cleared promptly after an apicoectomy. Pain that persists a full 4-5 months after the apicoectomy suggests that something is amiss, and the apico did nothing to address the problem. Additional waiting is unlikely to provide any benefit at this point-- what you need is a good diagnostician to clarify the source of your pain.
While I am not generally pessimistic in nature, I'm uncertain whether tooth #13 is to blame, but it has clearly been through the mill and must remain a suspect. Not all apicoectomies are successful, and there are other possible issues-- for example, a fractured root-- for which an apicoectomy would provide no benefit. The removal of the tooth would be a leap of faith in the absence of any clear diagnostic indicators; even if it is extracted, the pain might remain, but that would in itself be diagnostically significant.
The two other possible suspects that must be considered is (a) a persistence of your chronic sinusitis, or (b) chronic neuropathic pain. As a dentist, I am unqualified to provide specific guidance, other than to say the proper follow-up by an appropriate clinician is crucial. Chronic sinus disease is notorious for is refractory nature, and must be considered a prime suspect. In addition, there are neuropathic pain syndromes (e.g., chronic regional pain syndrome/reflex sympathetic dystrophy) that are known to be associated with prior surgical interventions, such as your sinus procedures and apicoectomy. Proper assessment should include a consultation with a neurologist.
It may even be worthy to consult with these medical specialists prior to taking any action on tooth #13, if only to possibly avert an unnecessary extraction.
Hope this helps...