That sounds like a solenoid that controls tranny shifts. Mine did the same thing and it took forever for the techs to figure it out. (This is before BC for me
)I wish I could remember what the part is exactly... I don't have my papers on the car with me. It was under warranty for me and it was very expensive as I recall. Hundreds of dollars. Sorry
But yes, if I shut the car off for about 20 secs to a minute, then it would run fine for a while... then jerk! I read on BC about it and you should try to get it fixed soon. Damage not likely in the short term, but don't let it go.