10V is too low. If the alternator is putting out 14V, it's probably not be the problem. I'm inclined to believe that you have a bad battery (see Spooler's remark above) or the charging system in not working properly. Get an OBD scan tool and see if any PIDs are showing up when the dash lights are on.
Battery voltage, with engine off, should be at least 12V. If it's dropping below that after sitting overnight, you have a bad battery or excessive parasitic load.
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