Yes on yout terrain your well pushing it. Your cassete should certainly not be moving around. The only way it should move is forward. A very simple way of knowing if your chain is worn is to either get a chain checker( £3.00) or measure with a ruler.
You simply put the chain checker in the chain and if you have over 0.75 of wear over the length you need a chain. The ruler method is simple also.
Put the ruler across the length of your chain and bear in mind we will be working in imperal. Put the zero on your ruler in line with the centre of one link. Then count across 8". If your chain is 8 1/16" its prob okay. If its 8 1/8" it needs replacing.
Toffer the way most people know that their chain is knackered is when you start upshifting to pedal easier and it starts skipping a sprocket or two. Most people then start adjusting things in frustration and eventually end up taking their steed down to the LBS to be told that it needs a new chain, cassete and the gears need re adjusting.
The chain is designed to wear before your sprockets.
So the minute your links start wearing and you get to the point whereby it needs changing, change it. This will prolong your cassete life immensly.
Dont wait until its too late or you will need a new cassete.
Heavier riders will have to change a chin earlier than a lighter rider. Also if your a cross chain merchant you will too have premature wear.
The chain should always be kept straight, not at an angle.
So I would take your cassete off Toffer and check the wheel. That may need a hub ?. If not put the new cassete on. If your not sure how to do that its a 2 min job with the chain whip and correct socket for the sprocket end cap.
Then put your chain on. Remember to check the length against the old one.