Model 12F702-0606-01 Code 9408095 Horsepower = 4
Has been in storage for years. Fired it up and lots of smoke came out of the muffler. It ran for a short while then died. Looked below the muffler and there was a small pool of dark oil. Checked the oil and it looks very clean. After several tries got the unit to fire up again. Same thing happened, lots of white smoke, oil dripping out of muffler, after several minutes I just shut it down.
Had been advised to let it run to burn off the oil, which I did while I proceeded to mow the back yard. Thought everything was okay. The next time I tried to start it, it fired up briefly, then it died rather rapidly. I tried it again, just a few sputters but nothing. I cannot even get it to fire now. Replaced the spark plug with one set to proper gap. Still nothing.
What can I do to troubleshoot this unit?
Also, does anyone know what the trick is to draining the oil?

Right , ok , well first of all the Briggs engine is a tough one so its very likely to be saveable. first thing i would do is drain and change the oil , it may look clean but it may also of degraded due to time. underneath the mower their should be a drain plug. alternatively you can suck the oil out of the dipstick tube with a cheap tool.
Secondly change the spark plug , a mower store will supply you a plug cheaply and some oil .. the correct oil is SAE 30 oil , any will do , but specific mower oil is best.. mineral! not synthetic.
And lastly.. this is actually most likely causing your problem.. change the petrol , tip the mower upside down get the old fuel out and fill with fresh.. petrol goes off and only really lasts 12 months anymore than that and its not going to be good enough to run the engine properly.

If you need any more help let me know , hope it works for u