At least as a suffixing verb in the present-day language, ～過ごす no longer has any "over-" meaning as ～過ぎる does, so the two are totally distinct grammars.
V 過ごす only means "V to miss something" or "V to dispense with something", thus:
- 寝過ぎる means you sleep too much (that causes headache, maybe), but;
- 寝過ごす means you sleep until it is too late, e.g. missed the station or school
- やり過ぎる means you overdo, but;
- やり過ごす means you do without dealing with something, that is, let go (past) or avert
- 見過ぎる means you over-look, or overdo looking/watching, but;
- 見過ごす means you overlook, in the sense failing to notice
Similarly, 食べ過ぎる is just "eating too much", but I have never heard of 食べ過ごす as a single verb, except for a collocation of 食べ + 過ごす "to pass time eating —".