B&s 31R707-0001-G1 help

I've got a mower I just picked up that easy running. Had stated smoking and eventually died. Motor is a B&s 31R707-0001-G1. Well ,I got it running,sort of. The sparkplug cap basically disintegrated when I removed it to check for spark. It would turn over but not want to fire. I got a new plug and adjusted and cleaned the coil and flywheel and got it to start. It was smoking really bad. White smoke,and smells like oil. After running it for a while when I shut it off it don't want to start again. The spark is real weak most of the time,now and then it has good spark and fires. But not after it's been going for a little while. I'm assuming the coil needs replacing. I did a compression test and got 130 on that. You can hear the oil sizzling in it after it's been run.(I mowed the yard with it). Then shut it off and back to the same issue.im new to working on mowers and thought I'd ask for some advice on my next steps. I plan to build a leak down tester.should I test the coil? Even the brand new spark plug is weak. Also I noticed it looks the the front of the plastic engine cover had been melted where it bolts to the front of the engine,so I'm assuming it's been pretty hot before. Any advice or questions would be appreciated. Thanks.