Do leftover successes from a roll after removing a Doom token get discarded or do they just start a new count of success? For example. I am rolling 12 dice in my Combat check and we need 6 successes to remove a token and currently we have 4. Lets say I roll 8 successes (real lucky right?) 2 of those 8 will add on to the other 4 to remove a Doom token since we have 6 now. Will those other remaining 6 successes be discarded or do they carry over and end up removing another token?

The 6 carry over. Think of it this way:

Yig—10 doom tokens
6 investigators

You need 60 successes to beat Yig. It doesn't matter what order they're dealt to him. The doom token removal just tracks your progress. There's no "resetting to zero" or anything complicated like that.