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It is not all the uncommon for this type of problem to occur, Ford has had many known issues with the radios and in most case replacement is going to be cheaper than having the radio rebuilt. The radio maintains power for memory even when the key is off.
do you think it is worth contacting Ford and asking them for help with this issue?
Buying a new radio is not in my budget, and I would immagine this is expensive.
what do you think?
Yes its usually more expensive to repair that replace . It certainly would not hurt to contact your dealer and if the want your return business they may help you. If not there are other options you can explorer.
You can often time buy good used ones on EBay and I have provide you with the link if you so choose to go this route.