The stock engine uses two oxygen sensors, one right off the turbo housing/exhaust manifold before the catalytic converter and a second downstream O2 (oxygen) sensor after the cat. They are both narrowband, which means they accurately measure oxygen levels to determine air/fuel ratio only at the stoichiometric point, or ~14.7:1. They can only tell the computer if the car is richer or leaner than that point.
The computer uses the upper O2 sensor to adjust fuel enrichment at idle and partial load cruising conditions that are around the 14.7:1 air-to-fuel ratio they can read. When you go into boost the computer runs calculations off pre-programmed tables to determine how much richer it needs to run for safety and to prevent knock. The computer learns if the car needs slightly more or less fuel at light load cruising conditions and can apply a corresponding adjustment to the fuel applied under boost or during wide-open throttle, but can't use the stock narrowband O2 sensor to verify.
The downstream O2 sensor after the cat isn't directly used for fuel calculations. It's primarily used to verify the efficiency the catalytic converter is operating and make sure it's in a safe operating range.
If the car has a third, aftermarket wideband sensor installed, it would be used to drive a gauge showing the actual air/fuel ratios since they can measure across a wide-range, generally from 8 or 10-to-1, so if you're watching a wideband showing 10:1 that may be as rich as it can read and the car could actually be running much richer than that. Note that some wideband O2 kits install their sensor in place of the upper stock O2 sensor, than provide a simulated narrowband sensor output from the wideband gauge/controller to feed the stock computer.
It sounds like you recently picked up the car and don't know much about the history. If that's the case, it's possible the previous owner pulled some or all parts off. If the computer has an aftermarket tune that doesn't match the hardware installed (i.e. expecting different fuel injectors, MAP (manifold absolute pressure) sensor, etc.) it won't run correctly.
When you say you adjust A/F and then it resets, what exactly are you adjusting? Some old turbo timers displayed A/F if tapped into an O2 sensor output, but most turbo timers (that keep the engine running a short period of time after it is shut off) would have nothing to do with air-fuel settings. Back when the cars were new and before the computers could be tuned (reprogrammed) owners used to install piggyback fuel controllers like the APEXi SAFC and others that slightly altered the MAP sensor reading to trick the computer into thinking there was more or less pressure in the intake manifold, which then allowed for less or greater fueling.
Eric H. - '04 Neon SRT-4 (eBlue/S2) - '92 Dodge Daytona IROC R/T
'91 Dodge Spirit R/T - '19 Pacifica Hybrid Limited - '14 CTS-V Wagon
'08 Cobalt SS Turbo - '14 Cruze Turbo Diesel - '02 SVT Focus (S/C)
'04 SVT Focus (EAP) - '07 Ram 1500 - '81 & '85 Chevy Citation X-11
'90 Pontiac Turbo Gran Prix (ASC/McLaren) - '93 Olds Achieva SCX W41