jessa is right, if it is that the crystal oscillator stops working is because of overheating, and it's caused
by the heavy duty fast processor and don't forget the new LTE chip, neither of them come with a miniatur fan. The little guy works as hard as a laptop. I remember few years back, one of the laptops with an AMD graphic chip used to blackout all of a sudden, without warning, and the cause was overheating of the AMD chip, because it did not have a cooling fan or bigger copper heat pipes, as a result the chip detaches itself from the m/board, boom nothing! poor consumers thought their LCD just went out.
Justin, I have a nice blue 5c like yours with exactly the same problems, everything works in my phone, except date and time. recently, I visited an apple tech., I told him the problem, without hesitation, he told me the logic board has to be replaced with $$$, my advise to you bud, replace the logic board, if it's fixed, get rid of it, i.e. sell, sell, sell it, that is what I will do, hope this will solve it. 'till then so long.