I did some research into this, as I needed to fix my rich-running engine. My engine did not have a good ground to it, and thus the O2 sensor (one wire) did not have a good ground.

What happens? Rich fuel mixture.

The computer doesn't see a good signal, so it runs rich to be on the safe side. That and after a prolonged period of time running rich, the sensor itself probably has gone bad. Rich fuel mixture tends to contaminate the sensor. There is a few number of sensors that have greater protection against contamination like the Bosch PosiLock.

I fixed the ground problem, but the engine still seems to be running rich, thus the conclusion of a bad O2 sensor. If I'm going to replace it, I want to replace it with a good one that will have a good connection and provide a proper signal even at idle.

Single-wire sensors tend to fluctuate in signal during idle due to temperature drop in the sensor itself. Single-wire sensors depend on exhaust temperatures to heat itself. All sensors need to be at 600 degress fahrenheit, or above, to send a good signal. Three, four, and five-wire sensors have a heater element within the sensor case. This insures the sensor is at optimum temperature regardless of how the engine is running, therefore provides a good signal to the computer.

Installing a three or four-wire sensor insures the computer receives a good signal and adjusts the fuel mixture accordingly. It also allows for one to tune the fuel mixture if used on a carbureted car or a car with a standalone engine management system. However, it is recommended that a five-wire, wideband O2 sensor be used for tuning with a standalone. After tuning has been done, you can switch back to a three or four-wire sensor. Wideband O2 sensors allow for more precise tuning of the engine, and they are a topic for another thread altogether.

Here is a presentation in what needs to be done.