"Shift to D2" Light

My automatic 92 Saturn had the "Shift to D2" light go on a few days
ago. Then the transmission started acting funny. While in "Drive" it
would accelerate very slowly as if starting from a high gear. When
shifting from "drive" to D3, D2 or reverse, the whole car would jump as
if the motor mounts are totally loose. As soon as I accelerated to
30MPH the "Shift to D2" Light would go off. After a day with this going
on the battery died just as I was pulling up to my driveway. I
replaced the battery and the car drove fine for one day. Now the same
problem started again. The "shift to D2" light is on, and the
transmission is acting funny again. What is going on?

1. What does transmission have to do with electrical power? I thought
it is purely mechanical.
2. I am thinking the problem might be that the Alternator is bad. But
I can't explain why the transmission is acting the way it is, and why
the "Battery Low" light did not go on.
Any insight on this problem would be greatly appreciated.

Thanks