Me and my brother both had Dell laptops and between the two of us we had three chargers break in less than 3 years. When our chargers were broken we had the exact same problem as you -- the charger would provide power but would not charge the battery.
But considering you just replaced the charger... not really sure what's going on. Find a friend with a Dell laptop and see how your charger(s) work on their computer?
Edit: I think there was a case where my brother bought a cheap adapter and it ended up not working, so just because it's new doesn't mean it will work necessarily... I encourage you to try another person's charger or have them try yours on their computer.
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No amount of planning will ever replace dumb luck.