Bad upstream or downstream O2 sensor? Car will run OK with a bad downstream one, but generally not forever. Though the upstream one is used for majority of the air/fuel mix for driving, both are used to adjust the overal fuel trims.
Couldn't hurt to pull the plugs and see how they look. Could be plug fouling - engine will run very poorly initially until it blows through the fouled plugs, then it runs great on the now cleared plugs. Could be a result of your oil consumption. If you see puffs of blue-white to gray-black smoke come out when you run into these poor acceleration issues, could be oil fouled plugs.
This family of engines are also exceptionally sensitive to electrical noise. A battery that is weak on reserve capacity and/or charging system that is not running 100% could cause this poor off-idle acceleration.
Oil consumption - sounds like fuel dilution. Engine will consume oil but it will be diluted with a higher percentage of gasoline (pretty normal for oil burners) - gasoline will make up the difference between the lost oil - so it "looks" like the car is not using any oil. Eventually you'll come to a certain concentration where the gasoline will rapidly cook-off and sudden oil seems to disappear. Adding a little bit of make up oil will help a lot - a small fraction of fresh oil added to old oil has been shown to boost the additive pack immensely.
For really bad oil consumers (loose 2-3 quarts over a 5000 mile oil change interval) - I keep a track of oil levels every time I drive away, or at least every fillup. Topping off every chance I get - even if the oil drops a couple of oz, I fill it up by that small fraction. About 500-1000 miles before I'm set to change the oil, I'll add a 1/4 to 1/2 quart of fresh oil - always ended up draining out a pretty much full crankcase by the time the oil change comes up.