I’m not entirely sure if this is canon, but the books and movies both seem to imply that Frodo suffers from chronic pain.
In the extended edition of FOTR, Gandalf asks Frodo how his shoulder is, to which Frodo responds “Better than it was.” I could be wrong, but to me that implies that the wound has pained him almost constantly since he regained consciousness, or at least he has had flareups. And he has flareups in both book and film canon.
Sam finds Frodo in the tower of Cirith Ungol after searching for more than a day. In the book, Sam asks Frodo if he can walk; Frodo says that he is not hurt but mentions a pain on the back of his neck, most likely where Shelob stung him. If it still hurts enough for him to bring it up more than a day after he was stung (keep in mind that Frodo seldom complains, so if he brings it up it must be bad), it is perfectly possible that that left him with chronic pain as well.
Having lost his finger in such a brutal way, it’s feasible that he experiences phantom pain, but that’s never addressed, so we don’t know for sure.
In the movies we get a few shots of blood on Frodo’s neck. Most likely the chain draws blood most of the time that he is in Mordor, which would be several days. Such long-term impact could have led to nerve and/or muscle damage, which would lead to pain, but again, the movies don’t address that, so it’s possible but not proven.
Also, the Tolkien wiki says that Frodo was in “continual pain” from his shoulder wound, so there could be other implications in book canon that I’ve overlooked.
So chronic pain could be another repercussion of the trauma he endured and another reason for having to leave Middle-earth.