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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.
 
I'm kind of confused on this subject. Some people tell me that keeping my MacBook connected to my A/C adaptor all day is a bad thing. I don't want to be wasting my battery by unplugging it while its sitting on my desk inches from my a/c adaptor.

I want to maximize my battery, and from what I've heard if I continue to keep it plugged in all the time and let it sit at 100% fully charged, I'm doing more harm then good?

You can leave it plugged in and 100 percent charged all the time. Just once a month, unplug it and use it until the battery runs out completely and the Mac turns itself off, then plug it in again and charge it until the battery is full. The batteries want to be completely emptied once a month.
 
Thanks for all the response guys, but like that one poster wrote, "batteries are [indeed] fickle things". So I'm still around 680s in terms of battery cycle but my battery health is around 90%-95%, so I'm wondering if it's the cord.

Ok, you have to wriggle the power cord around to get it to charge as you say in your first post, and you're asking if the problem is with the cord? I'm going to go with "very probably, yes, either that or the socket. The fact that playing with the cord helps is a bit of a clue".
 
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