When plugged in, does the indicator light on the laptop showing it's plugged in stay on the entire time, or does it go off if you move the connector tip inside the DC jack?

If it goes off, then you either need a new power adapter (easy to test with another similar adapter with correct voltage input) or your DC jack/jack cable/DC board needs replacing, whichever your laptop has.

If it stays on then it's possible that you have a clogged heatsink causing your laptop processor to overheat. Laptops are supposed to shut themselves off if the processor gets too hot to prevent permanent damage. What brand and model is your laptop?