I have no advice to offer, but I hope you can figure out what's wrong and get it remedied.
Ten years ago I was looking to buy a .22 LR for kids with a good trigger for as little as possible. Ended up buying a Savage MKII Youth, serial no. 161xxxx. I also have only shot a couple hundred rounds through it, but it has always cycled flawlessly. I usually just used the (formerly) cheap Walmart Federal .22's. I never did any serious target shooting on paper, but the one time I hunted prairie dogs with it, it seemed to especially like CCI Velocitors, hitting accurately out to 100 yards with open sights. I love the adjustable Accu-Trigger and the tiny front bead sight. It's a great little shooter.
You were asking for advice about a single shot rifle - have you tried loading your Savage single shot (without the magazine)? I tried mine and it's a bit tricky to load, but doable, and extracts just fine.