Does sin befall me for delaying the Aqiqah beyond its seventh day, and is it permissible to donate its monetary value instead of slaughtering?

Delaying the 'Aqiqah is not a sin because the preponderant opinion is that it is a Sunnah and not obligatory, and you had an excuse for the delay. You can perform the 'Aqiqah now, and it is more cautious to adopt the view that its time does not expire at all. The 'Aqiqah is a ritual for which it is not permissible to give its monetary value; rather, it must be performed by slaughtering an animal, as the Prophet, peace and blessings be upon him, did.