Need some help with a Dodge Neon... Trans shifter solenoid, limp mode..

[420sx]

I just picked up a 2001 Neon as a project/investment. The PO told me it had a blown trans. Upon doing a bit of reading and researching, it seems more likely that the TSS is bad.

Symptoms include: wont rev past 4k in neutral, or 3k in gear (limp mode?). It wont shift into third gear. The revs rise slower than they should- it sounds like the motor is under heavy load when I rev it in neutral. Also (probably unrelated) but the battery goes dead after sitting for a day. BRAND new battery.

The thing thats stumping me is, theres no check engine light. I would assume if the car was in limp mode, itd have a CEL. I'd like to be able to confirm my suspicions before I go out and spend $155 on a TSS. It seems to have most of the symptoms of this problem, but id really be kicking myself if i replaced the part and it didnt solve the problem. Can anyone offer any advice on the matter?