possible O2 problem?

[silverkwik1]

I just bought my 98 gsx and have had trouble with fuel cut at around 5500 rpms. Since then It seems to be getting worse. It will run "fine" until out of nowhere it will start to cut out. My last time fueling i got about 16 mpg and it will run out of gas at a little over a 1/4 tank with it only consuming about 13 or 14 gallons of gas. I had some CE lights checked. The first one was a
"fuel trim malfunction(bank 1)"
and the other was
"clsd loop fuel Ctrl, insufficient coolant temp"
Looking at other threads it seems like the O2 sensor might be the problem, but is there any other possibilities? Also, why would it run out of gas so early, is that the O2 sensor?
Any help would be greatly appreciated, thanks

 

[silverkwik1]

Today I decided to unplug the front O2 sensor completely and the car ran for about 4 seconds then died. This is the same thing that has happened with the O2 sensor hooked up. Should it make a differance on a perfect running car if you have the O2 sensor plugged in or not?