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The dates are only updated when the payment is completed. If you allow for offline payment (e.g. cheque), then the dates are not updated yet when the invoice email is sent.
In that case, the subject of the message should be invoice rather than receipt.
As a workaround, you might want to remove the dates from the template or create tokens but those template ...
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You may want to look at the Membership Extra extension, which will let you prevent people from renewing until, say, 30 days before their current membership end date.
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