Here are your parts
Parts for Frigidaire PLCF489ACB Range - AppliancePartsPros.com
Here is the wiring diagram
Since the element came on by itself I would say that your clock/timer (item 20 in the Backguard Parts section is toast.
The electronics that control the broil relay are holding it energized all the time.
The F3 code is for an open oven sensor but this may just have been because the oven was overly hot.