Did it stop smoking?
If so, it was probably just some "coating" burning off the exhaust system internals of a NEW engine.

I ran the washer for maybe 20 total (stopping once after noticing the smoke. I started it again and it didn't smoke for a couple of minutes but then started again. I checked the dip stick wondering if I had in fact unknowingly added too much oil, and it did seem too full. I tilted the engine and pored out the oil and it was much darker that I felt it should have been. Maybe I did screw up. I just don't know and not sure if I should refill it and try again. I understand you can blow the head gasket by overfilling the oil. Or maybe I should just take it back to Home Depot for them to send it off for analysis. I just can't believe how this could have happened.

Yes, I would refill to the proper level- 10W30 or SAE 30. 2) Check the air filter element for signs that it may have been saturated w/ fuel or oil. Replace if this is so. 3) This engine doesn't have a head gasket- but if you have done the above and it still smokes after 5 minutes of running- back to agent orange it should go.

It can be difficult checking the oil on some engines shortly after shutting them off.
However- You can check it after it's sat for a couple hours and get an accurate reading.

You seem a bit to eager to pour in/out oil without really knowing what the true level is! SLOW DOWN---

Thanks for the info and advice. Yes I was probably not careful enough in monitoring the oil level. I just assumed since the container only held 18 oz and the engine capacity was 20oz there should not be a problem. However I remember in the Army the definition of "assumption". Thanks again.