1993 buick lesabre shut off while driving,battery was dead then bought a new battery drove well for 3 weeks then shut off while driving again had to jumpstart it.

I think the mechanic could very well be right in this case

Make sure the trans cooler is getting a good air flow. and a trans flush is always good, and most oil change shops do it for around $89+

Exhaust Converter over 12 years old & blocked ?

Gas Filter Clogged ?

TPS Sensor adjusted correctly-if adjustable?

Fuel Pump may have failed in some way,check pressure

You have something draining your battery when the vehicle is off. Connect an ammeter to see what amperage is drawn when the vehicle is at rest and off (with the doors closed so that you don't read the draew from the dome/courtesy lights). Begin pulling and replacing fuses one at a time to determine which circuit is causing the abnormal current flow then troubleshoot that circuit. Many times the fault can be traced to a glove box, trunk or underhood light that does not shut off.

Check to see if it is only a bad head gasget. If you bought it from a dealer take it back.

A relay or the wires connector to the motor needs to be cleaned or replaced.

sometime the ignition switch will cause this problem, a condinuity test is recomended to ensure the contacts are not engaged when key position is off.

The locks don't go thru the ignition switch. Check the battery connections for corrosion or looseness.

check under the hood of the car instead of within the chasis for the headlight fuses.

it possable you need new cable i dont know your car.
The battery cable can make so you car will not get the power need to start or turn on lights.