No need to swap out the pickup if it's just a tuning issue.
You could lower the Low E side of the pickup with the adjustment screws to see if that helps,
(if it's real close it
can pull on the strings too much....I'm guessing it's something else though).
Has your intonation been set when you last changed strings?
It could be the nut slot is tight on the low E.
Pics are a plus !
(did I mention we like pictures?

)
welcome aboard,
elwood