My rear wheel on my mountain bike keeps getting out of place latterally when I climb?

I doubt your rear wheel movement is down to stripped threads on your quick release - that seems nearly impossible to achieve. The threads are well protected and the lever action of the quick release mechanism means you don't actually have to "work" the threads. The problem is more likely to be the cones and/ or bearings in your rear hub and/or your bottom bracket. ie the moving parts that wear out. You have to replace and maintain moving parts - they don't last for ever. If your back wheel moves from side to side even a fraction when you grab it with your hand and shake it - you need to tighten your cones (with a cone spanner) this is necessary maintenance - as your bearings wear the gap between the bearing and the race will cause more wear until your tire starts knocking the frame. It's the same deal with the bottom bracket but not much you can do to maintain a BB - just have to replace them. You should not be able to move your cranks from side to side they should be nice and stiff. Chances are if the bearings in your hub are going, your bottom bracket may be on the way out too. Nearly all bikes have a shelf life of not more than two years if they are "ridden" by the owners.

Try tightening the skewers first.

That's scary. Since you said it looks worn out, you better replace those skewers right away.