Ok. That's the governor pressure solenoid. It's the part that actually allows fluid pressure into the governor circuit based on the signal from the governor transducer (pressure sensor). The transducer is the one that points forward. It's very likely your problem. Sometimes it will set a code, sometimes it won't. In your case it sounds like it is setting a code since disconnecting the battery makes a difference. It would be best to get a scan tool or code reader on it and see what codes are stored before spending any more money, but it does sound like you've probably got a transducer issue.

Ok great. That does sound like a transducer problem. There's an outside chance of a shorted oxygen sensor contributing to that code though. The sensors are on the same 5v reference circuit and an internally shorted oxygen sensor can cause this problem without setting an oxygen sensor code. Slide under it and check out the wiring to both sensors. If it's ok, unplug both sensors. If the problem is still there replace the transducer. If you can get definite results from unplugging one of the oxygen sensors replace it.

Yes. If you don't have California emissions you just have two oxygen sensors. They are both mounted in the exhaust pipe. (If you have California they are still in the pipes but there will be more sensors) In the Federal emissions model you will have one sensor either in the pipe just in front of the catalytic converter or screwed right into the front of the converter. The other sensor will be on the pipe after the converter.