2003 Hyundai Elantra When driving the

Have your transmission filter changed and your transmission fluid flushed.

Sounds like the idle sensor needs replacing

Did you check the transmission fluid level? If necessary, add fluid. However you probably have a more serious problem. Take the car to a qualified transmission mechanic.

The vehicle speed sensor is probably faulty. Have it checked. it's mounted on the side of the tranny.

is the shifting due to the transmision? or the engine? i had a friend who's car did something simular, diffrent car though and a older model. turns out, it indeed was the fuel pump, when they thought it was the tranny. try some Lucas for the transmission. if that dont help then i would let the mechanic drive it.

Does the Check Engine light come ON when this happens? Whenever the "Check Engine" light comes on, there is always a DTC (Diagnostic Trouble Code) stored in the ECM (Engine Control Module). You'll need an OBDII (On-Board-Diagnostics) scanner to retrieve any codes recorded/set in ECM. Once you do this and let us know what, if any codes are set, then we can go from there.

DTC P0734 is "4th gear incorrect ratio" - that's your torque converter 'LOCK UP Clutch" It's worn out (or you're using the wrong Trans Fluid required by Hyundai - SPIII OR Both). The P1624 is simply the Trans Control Unit (computer) communicating to the Engine Control Module (ECM computer) MIL - on request signal to ECM = asking it to turn on "Check Engine" light. MIL = Malfunction Indicator Lamp. When the torque converter 'lock up" clutch slips or doesn't engage, the trans automatically returns to 3rd gear and stays there (it's called 'fail safe' mode) until you cycle the ignition key, then everything resets until the malfunction occurs again.
You'll need to have an AT (Automatic Transaxel) tester hooked up to verify this. Hyundai has a Service Bulletin specifically for "gear incorrect ratio' codes. If when you changed the trans fluid, was it done at a Hyundai Service Center? If not, someone may have installed the wrong fluid.

Go to www.hmaservice.com and register vehicle by VIN #. Then you will have access to shop manuals, service bulletins, wiring diagrams, etc.

Have u checked the fluid level in the transmission do this only when trans is hot, if fluid level is OK have the trans fluid and filter changed