Not the VIN, but the trans case has a number on the driver side above the pan.While you're under there take a look at the shape of the pan. Compare to the gasket shape in the pic on this page:
That'll help narrow it down.
The sluggish behavior could be a need for service. Fluid, filter, and band adjustment. Older Torqueflites also eventually experience a situation where the torque converter will drain fluid back into the transmission as it sits, particularly overnight. When you start up, in Park, the pump doesn't actually pump. Go figure. Anyway, when you start up, slip it into Neutral for just a couple seconds. That'll let the converter get some fluid into it. Then drop it in gear. If response is better, there you go. This can be addressed with a valve body kit. B&M, TransGo, Superior, Sonnax....etc. Not only will they fix this, but they also allow you to check trans fluid in Park, instead of Neutral..... nice little safety perk there.