Sounds to me like either an earthing problem, or a relay. I would chase the starter relay first (if is is actually staying on after you release the key then this would be the most obvious)
Also, I dunno about the actual voltage levels at the solenoid, but AFAIK after the key is moved from the start position back to the on position, there should be NO power at the solenoid. Once again, this points to the relay.
12.8 volts at the battery seems a little low to me - it should be at about 14 volts (well, mine always are, regardless of the car) Also, The second wire from the solenoid (-) is only required if the starter casing doesn't provide a solid earth back to the chassis. If the - terminal is soldered to the casing then you can assume it is internally earthed? (hopefully someone else can confirm/deny this theory)
Also check your earthing strap to the engine. If you have a dodgy earth in the engine bay somewhere it may be trying to earth back through the solenoid wiring.
Sorry I don't have any sure fire fixes to offer.