Well thanks for everyone's tips and ideas. I've been to my local Grainger. They sold me a capacitor for my older table saw motor, which now runs fine and I am at least opperational for the moment.
As for my newer craigslist aquisition, it is still a boat anchor for now. I have opened up the motor, and fiddled with what I believe is the cetrifugal actuator switch (a couple "blade like" contacts attached to springs and mounted on the main drive shaft). It seems to be moving freely. Spray'd some wind and some cleaner in there to make sure. Put it all back together and fired it up with the old Capacitor wired in. Fired right up, ran strong, and this time the Capacitor started smoking almost immediately. Shut it off quickly.
Grainger did not have a Capacitor for that motor. At least one that would fit in the cylindrical, stamped housing that bolts to the side of the motor and holds the capacitor. They had one that would work with similar specs, but the physical size was much bigger. They suggested that I go to a local motor shop to find the odd sized capacitor. Luckily, I got out of work early last Friday, and did just that. They told me that the only capacitors they had, would be the same as Graingers, and that it appeared that mine was a Japanese Capacitor, probably from a Japanese motor. They said they could bolt a new housing to the side of the motor to fit one of their bigger capacitors if I bring the motor in. I'm gonna have one more go at finding the exact replacement online and see if a new cap solves the problem before bringing it in to the motor shop.
I'll update later after that mission plays out. Thanks again.