Oh my. I think your second pic nailed it. Pull the engine out of the case (not as hard as it seems) and get every bit of that sawdust out of there, your engine is not getting any cooling air from the fan right now and the steam or smoke is probably evaporating oil. You have a big fire hazard going on there too, especially if some fuel find its way down there. My dad lost a nice old Homelite Super XL this way.
If the 303 is like my 3000, it would oil a 32" bar if you could find one with a small Echo mount. Meaning, it grows a shell of oily sawdust like an old Detroit Diesel coats itself in oil and the oily dust will build up inside as you can see. I don't know any way to prevent it other than adjust the oiler to the minimum required and periodically clean out the sawdust. BTW, if your dad ever wants to get rid of the 303 let me know, I wouldn't mind having one.