The problem you describe is a common one. I've experienced it with my 600 and have read posts by others describing the same issues.
When it was new my 600 extracted and fed consistenly, without having to briskly cycle the bolt. After maybe 4000 rounds it started acting up - perhaps from wear or a bent part? The next round would not be pushed up high enough to chamber freely. I'd have to fiddle to get it to chamber. Maybe it's due to a worn magazine spring, so the follower thingy does not exert enough upward pressure. Am guessing here.
I never did figure out what the problem was as I only did a cursory examination than retired it to the safe.
The Cooey was replaced by an Annie which solved the extraction/feed problem quite satisfactorily.
I hope others here know what to do as I'd love to get the old girl back in action. Like most Cooey 22s, she was a fine shooter even with the coarse irons she came with.