The symptoms sound like a combination of lack of lubrication, and perhaps lack of fuel (if the speed actually increases before it stops). I don't see a loose piston causing the speed increase, and unless the lubrication was bad, I don't think it is likely to cause seizure either, unless it was working very hard.
Has your old oil separated out from the petrol? If so, oil drops could be blocking the main jet and causing fuel starvation, while doses of straight petrol could be causing seizure. It isn't a perfect fit for the symptoms - I've never had full separation with any kind of 2 stroke oil, though I've had it with engine oil used to mix 2 stroke fuel.
Otherwise we'd be looking at something a bit obscure, such as broken or jammed rings causing intermittent seizure, and the speeding up before stopping could be in your imagination, due to the governor opening the throttle when it starts to seize, and making louder exhaust noise.
