Jim,
I would say you have a fairly typical handi. My 223 does the very same thing. When I first got it, it shot 8-inch groups, always strung out vertically with the first shot being the highest. Took me a while to figure out that if I stopped cleaning the barrel, the groups would tighten up. It came with a very rough bore, and I continued to get heavy copper fouling until I had put >200 rounds thru it along with a lot of bore paste. Its now got 400 rounds and it doesn't copper foul so badly. First few shots after cleaning still go high though. This is what works for me. After a thorough cleaning including copper removal (and usually a little bore paste), I take it out and shoot a 5-shot "group". Shot 1 is several inches high and #2 is 1-2 inches high. Shots 3-5 generally make a decent group, and I consider the gun ready to shoot with. After shooting, I either put it away as is, or if its going to sit a while I run a single patch of Hoppes #9 to remove powder, a single dry patch, and a patch of CLP to keep it from rusting, but I make no effort to remove copper. I continue like this until accuracy falls off (usually >50 rounds). Then I give a thorough cleaning including Sweets and bore paste, take the 5 fouling shots, and I'm good to go again for a while. This is just what has worked for me. Enjoy your gun. Once you learn how to make it work, a handi rifle can become addicting.
Duane