I haven't yet picked up my actual travel ticket, but if it says "Manchester Piccadilly" as my destination, must I stay on the train rather than alight at Manchester Oxford Road, as I would now prefer? And if the ticket actually says "Manchester Stations", will that make a difference? Thanks!
Advance tickets are only valid in accordance with the itinerary shown on the ticket. Accordingly, you must use it to the station you've booked - and if your train comes to Manchester Piccadilly from the south, you would be overtravelling by continuing to Oxford Road. This could constitute an offence, depending on the circumstances.
However, if your train comes into Piccadilly from the north, then you are merely performing a break of journey where not permitted. The penalty for this is to have to pay the excess to the cheapest valid ticket that permits break of journey, i.e. the Off-Peak/Anytime (Day) Single (depending on time of day).
In practice it's unlikely there would be an issue either way, as the barriers do not read the reservations/itinerary but simply look at the ticket origin/destination. But there are almost always Northern revenue protection staff at both Oxford Road - so if your ticket is rejected by the barriers for any reason, and the barrier staff don't let you through, you may enter the web of Northern's hard-line revenue protection processes.
The safest option would be able to obtain a voucher for your booking under the Book with Confidence scheme*, and then use this rebook to Oxford Road. Of course if you bought your Advance a while ago, the new ticket might cost a lot more...
*NB some retailers such as TrainSplit do it slightly differently, giving you a refund of your old booking when you make a new one for the same journey - so this wouldn't work in that case.