Why isn’t my car shifting out of 1st when driving?

The TCC, or torque converter clutch, is used to lock the torque converter to the output shaft. Generally it's only used on the freeway to give you 1 or 2% better gas mileage on highway driving. In most cases, you generally wouldn't even notice if the TCC solenoid has failed. So the error code you're getting shouldn't have any effect on whether you can get out of first gear or not.

The interesting thing here is the traction control light coming on. What we need is to know what conditions on your car will cause that warning light and work on diagnosing that first and hopefully the resolution to that will be the answer to the transmission problem.

Poster - do you have or have access to an OBD-II code reader? It may give you some information on what's going on with your car even if the check engine light isn't on. If you don't have one, most auto parts stores will scan your car for codes for free, at least here in the US.