New battery causing new fault codes?

It looks like you've provided a list of trouble codes for your 2001 Ford Taurus. Each code corresponds to a specific issue or sensor reading that the vehicle's onboard diagnostics system has detected. Here's what each code means:

P0750 - Shift Solenoid 'A' Malfunction:
This code indicates a problem with the shift solenoid 'A' circuit. The transmission shift solenoids control the flow of hydraulic fluid to change gears. This code could be caused by a faulty solenoid, wiring issue, or transmission control module problem.

P0743 - Torque Converter Clutch Circuit Electrical:
This code indicates a problem with the torque converter clutch circuit. The torque converter clutch engages to create a direct link between the engine and transmission for better efficiency. This code could be due to a malfunctioning torque converter clutch solenoid, wiring issue, or internal transmission problem.

P0760 - Shift Solenoid 'C' Malfunction:
Similar to P0750, this code indicates a problem with shift solenoid 'C'. It suggests an issue with the transmission's hydraulic control circuit for shifting. The same potential causes as mentioned earlier apply.

P1451 - EVAP Control System Canister Vent Solenoid Circuit Malfunction:
This code indicates a problem with the evaporative emissions (EVAP) control system's canister vent solenoid circuit. The EVAP system prevents fuel vapors from escaping into the atmosphere. This code could be due to a faulty solenoid, wiring issue, or a problem with the EVAP system.

P1747 - Electronic Pressure Control Solenoid 'A' Short Circuit:
This code indicates a short circuit in the electronic pressure control solenoid 'A' circuit. The electronic pressure control solenoid is responsible for controlling transmission line pressure. This code could be due to a short circuit in the solenoid circuit or a problem with the transmission control module.

P0135 - O2 Sensor Heater Circuit Malfunction (Bank 1, Sensor 1):
This code indicates a problem with the oxygen sensor (O2) heater circuit in Bank 1, Sensor 1. The O2 sensor helps measure the air-fuel ratio in the exhaust gases. A malfunctioning heater circuit could affect sensor operation. This could be due to a faulty sensor, wiring issue, or problem with the engine control module.

So, I would clear the codes and see if they come back. It could be a glitch due to connecting the battery. If they do come back please start a new thread for each code.

Thank you very much Sir for taking the time to respond! Your explanation of these fault codes in layman's terms is more than I was expecting!
I retired recently after 47 years in the collision repair industry. Most of those years were spent building hard hits which usually require some sort of frame/suspension repair/replacement, but anything more than replacing a crushed pig-tail, splicing a cut wire, or checking for blown fuses, my electrical and diagnostic skills are very limited, to say the least.
I cleared the codes and unfortunately, they came back. As I stated in my original post, none of these codes appeared before my battery mishap.
So, I guess my question to you is, which code should I address first?
The car probably isn't worth the money I'm sinking into it. I recently replaced all 4 struts, tie rods, control arm bushings, brakes and rotors, plugs, wires, fuel pump, all the fluids, and 4 new tires. Oh, and that $150.00 battery. The thing has good compression, been we'll maintain since I bought it in 08 and all of all, it has been a very reliable ride up until now.
Once again, I want to thank you for your help!

If all of these codes came up at once after the battery replacement, I would say the voltage surge took out the PCM, you can get a preprogrammed unit by searching Google or Ebay. Here is a guide to help walk you through the repair and I have included instructions in the images below for your car:

https://www.2carpros.com/articles/how-to-replace-a-vehicles-main-engine-computer-pcm-ecm-ecu

Check out the images (below). Please let us know what happens.

Thank you! I will post my progress as it happens.

You are welcome, please do!