Small engine repair help

So I have a cheapo 4 cycle weed eater/trimmer. I replaced it about a year ago when it died, but I wanted to get it running again to use as a backup or give to someone who needs it. It would bog down when I throttled it, but would get to full power; then hard to start, and finally no start. I replaced the spark plug, fuel line, primer bulb, and fuel filter. For whatever reason, there is no air filter. I adjusted the valves. Checked the exhaust for excess carbon and looked fine. I took the carb apart and cleaned it, to my knowledge they don't make a kit for it. I soaked the new fuel filter in sea foam for a few minutes and put a cap full in the first tank of gas. Primed, choked, and it fired right up. It idles fine and hums at full throttle, but still bogs down if I try to go from idle to full throttle to quickly. I tinkered with the carb settings and it's a little better, but still bogs if I gun it. A little white smoke is coming from the carb, and also carb is spitting what looks like a oil/fuel mix. I use non ethanol gas, but not until it was probably a year or so old.

I don't really know what I'm doing, just tinkering around. It seems like I'm close, so just wondering if someone on here more knowledgeable than me can help. And also wondering how running another tank of gas without the sea foam will effect the new carb settings, and if the sea foam may be the cause of the white smoke and oil coming from the carb. Thanks for any help.