Transmission Control Solenoid?

Whenever my car is placed in drive (auto trans.), It revs, but will not accelerate. It may as well be in neutral. However, if I manually shift the vehicle down to first, then up to second, then third, it will still accelerate like normal, I just have to shift myself, and 3rd is the limit (plus it turns off the traction control). Reverse works as well.

Some co-workers of mine who are a bit more mechanically savvy than me informed me that what could be a good reason for this is that the transmission control solenoid is kaput, so it may be worth replacing before putting the investment into a new tranny.

I drained out the transmission fluid, and it was on the dark side (think cranberry sauce at thanksgiving, minus the chunks). But was unable to find the TCS. I'm thinking it's on the side of the transmission, rather than underneath inside the pan.

So, my question really boils down to 2 things:

A.) Does it seem like a TCS problem, or is this one of those things where I need to suck it up and start looking for a used tranny?

B.) If it is the TCS, where would I be able to find it on a 2001 Buick Century to replace?

Best regards,
Max