Well I would drain the battery if I could, but like I said, my computer shuts down every time I unplug the charger.

I don't see any information on the battery, just that it's performing normally and is fully charged.

But we ordered it from the Dell website....

Click on the battery icon in power meter in power options through the control panel. Does the info there looks like bogus / or listed with empty fields.

Laptop batteries suddenly dying not long after purchase can be "refurbished" by a factory reset without a battery cell replacement.

Drain the battery fully and do a simple laptop battery test to have an impression of the Full Charge Capacity.

Check with the supplier if capacity is way below the specified capacity.

Actually, something weird is that my battery meter says that it is 100% charged, charging. Meaning it's still charging? Should I just let it sit for a few days?

The laptop battery LEDS (if applicable) are a good lead to it's state of charge.
The leds are directly connected to the laptop battery gauge chip.
Meaning its well possible the battery cells are ok.

The water test would be to insert it in a similar laptop. If its ok there, there's a problem with the MOBO.

Try NHC (Notebook Hardware Control), it shows a bit more info about the laptop battery.

I probably should have mentioned that I have another battery that is reaching the end of its life, which is why I got a new one. But the old one charges just fine and holds about a 20 minute charge (which obviously isnt good but it means that my computer cant be at fault, right?)

If i was in your shoes i'd rush back to the supplier.