Do I need a new chain & sprockets?

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Joined
Aug 2, 2017
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Bike
Kawasaki KLR 650
After replacing both front and rear inner tubes on my KLR 650, and when adjusting the chain slack, I noticed that the slack did not stay the same as I turned the wheel. I got advice that its time to replace the chain and sprockets as the sprockets wear down to an oval shape over time, thus causing the increasing/decreasing slack.
How true is this? Do I need a  new chain & sprockets?

On a side note, the front end of the bike feels a bit wobbly when riding. When I changed the front tube, I lacked the common sense to mark the rim position on the tyre. Could the wobblyness be from an unbalanced front wheel?
 
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