Battery, O2 sensor, transmission miss fire

The sensor is probably one of the O2 sensors. The odor could be too much fuel or not enough.
What is the trouble code stored in the computer ?

run fault codes and check for HO2S and O2 sensors problems . Check fuel pressure and injectors for operation.

could be ether of these. try unplugging the front o2 sensor see if it runs better. if so its that o2 try unplugging the rear o2 if its better its that o2. by unplugging it the computer just selects an average input of a good o2. could also be many other things but start with that and I would change the fuel filter maybe its clogged

yes, it should run, but you'll have an engine light on because your downstream O2 sensor will think there's a miss fire.

The DTC (Data Trouble Code) P0135 is for the O2 sensor Heater Circuit Bank 1 Sensor 1. Bank 1 will always be the cylinder bank with the number one cylinder and spark plug, and the number one O2 sensor will be located before the catalytic converter and usually at the rear of the exhaust manifold or just after it. According to the firing order diagram bank one is the cylinder bank that is closest to the fire-wall and there should be an O2 sensor located there around the rear of the exhaust manifold or just after it. If the engine is a 4 cylinder then it will be the O2 sensor before or up-stream of the catalytic converter. Check for any damaged wiring or loose wire connector at the O2 sensor and if they are alright then replace the O2 sensor and then re-set the check engine light. (The auto part store should re-set the check engine light for you if you purchase the O2 sensor from them, and be certain that it is the correct and assembled O2 sensor for your vehicle and not a put together universal O2 sensor)

If it is a 4 cylinder it will be on the exhaust manifold at the front, a V6 is also mounted at the front, but lower down. If you are getting an O2 sensor code, change the fuel filter first. O2 sensor seldom go bad.

if its a 5.7 change your distributor cap and rotor and problem will be solved guarantee it

If it won't go into overdrive a speed sensor within the transmission may not be sending the shift signal to the transmission. Check fluid levels to make sure that's not causing the problem. Also, check that the fluid is red color and clean.

Transmission probably needs to be serviced at a reputable shop if you want this problem fixed.

Here is a diagram of your motor with the firing order:

You will need to replace the valve body in the transmission.