I think if you have 120 and 240 it must have a shared neutral unless there are 2 completely separate windings and a meter should show you that (ohm scale). My bet is you have 120-0-120 just like your house. Have you tried measuring between the hots of the two 5-15s? Is that is 240? Is there continuity between those neutrals? (more evidence that it is likely center tapped). The open question is if the neutral is bonded.
I used to repair those generators. No, I do not have any schematics. However Century Powermate built most of the store branded equipment. Therefore Snap-on, Sears, Montgomery-Wards, Coleman, Schauer, any of the automotive part store chains branded generators, welders, battery chargers, etc are all built by Century Powermate. The only differences are typically sheet metal and paint.
If you can find a schematic for a similar sized private label brand generator, it will probably be the same schematic.