[Backtobasics]

The oil burning is the culprit. Oil is being burned, and it is going out the exhaust, and coating the catalyst, and the oxygen sensor. Converters do not ususally FAIL for no reason. Oil, coolant, or excess or lean fuel conditions send contaminants into the system.

This is probably not going to get better. You can replace them both, but the burning oil is going to continue to contaminate the converter.

Possible short term is to use a spacer that pulls the oxygen sensor back a bit out of the direct exhaust flow, to minimize the oil coating. The converter is going to continue to fail. Best case is that the smokey exhaust has coated the honeycomb inside, and the exhaust merely flows through, unobstructed. It is possible it could start to build up on the honey comb material and impede exhaust flow, making the engine work harder to get the exhaust out.

You can consider fixing the problem (another engine, or engine rebuild) or cut out or gut the catalytic converter (it is not doing it's job due to contamination). Gut or cut out is technically against the rules, but if it gets clogged you are going to lose efficiency and MPG. Unless you are willing to fix the basic engine problem, eliminating the converter makes some sense as it allows you to be as efficient as possible, consider the burning oil situation.