Should I be worried?
No need to worry, but at some point you might need to take some action.
You are saying that the connector doesn't work sometimes. That happened to me as well; if you look at the connector, there are four small pins inside it and a different sized pin in the middle. These pins are mounted on tiny springs that you cannot see, so if you press them in, they come out again. That can stop working; it is very hard to see unless you compare it to a new working connector, then it is quite obvious. As long as wiggling on the cable helps, you are fine. At some point it might stop working; I bought a new charger at that point (seems impossible to find the cable on its own).
Every battery will eventually wear out. It depends on the number of charge cycles, how well you looked after the battery, and probably a factor of good luck or bad luck. 683 cycles is quite a lot, so your battery might not last as long as it used to. Again, nothing to worry about. If you are happy with how long it lasts, fine. If you are not happy with it, you will eventually have to buy a new battery.