Engine oil burning mystery

The engine is in a 1990s Honda H5013 tractor. This is a small tractor with a 13 HP air cooled engine. The tractor sat for 5 years in my brother's yard without being started or moved . My son went down to California, near Salinas, and picked it up. After only a little work, like changing the oil, draining the old gas from the tank, and putting in a battery the thing started up and ran pretty well. It burned oil when first started but then this stopped. Now that it's up here in WA I cleaned the carb real well, and my son adjusted the valves, were only a little loose, only .001". The thing starts up easy and runs well but still burns oil intermittently. When my son adjusted the valves we saw no oil pooling under the valve cover so I don't think oil is being sucked past the intake valve stem. But I could be wrong. I checked the oil level and I don't think it is over filled. I am checking the oil level according to the dirctions in the manual. Yesterday I drove it down the hill to my neighbor's house and it really started smoking. Then as I drove up his driveway the smoking lessened. The downhill slope and uphill slope were not very steep slopes. The smoking is for sure oil smoke, white and thick in this case, it's not the black smoke you see from an overly rich gas mixture. Like I said, it's intermittent. I will be driving on a level slope and the engine will start pumping out smoke for a while and then stop. Sometimes there is a lot of smoke and sometimes just a little and sometimes none at all. Aside from the smoke the tractor is a great little machine, 4 wheel drive, 4 wheel steering, locking rear diff, and is the one I want to add hydraulic blade swiveling to. Since the blade only descends by gravity I want to add a cylinder to push the blade down too. Any thoughts? Thanks, Eric