So the DPFE valve isn't supposed to melt. Yours melted because hot exhaust gas got into it.

That happened because the orifice in the EGR pipe is clogged with carbon.

Your DPFE sensor has 2  tubes going to it and they're connected to the EGR pipe. (that goes down to the exhaust manifold)

In between, where each is connected to the EGR pipe, inside the EGR pipe, there's an orifice.

The DPFE sensor measures pressure differences on each side of that orifice in the EGR pipe.

Your orifice is clogged with carbon. That allowed hot exhaust into your DPFE sensor and melted it and it will melt your new DPFE sensor if you don't replace the EGR pipe. It's important to also change those tubes because the same exhaust gases that melted your DPFE sensor probably clogged those tubes.

It's possible that your "no start" is also being caused by carbon in the EGR system.

It's possible that your EGR valve is stuck open. (because of carbon/soot).

The good news is that you seldom need to replace an EGR valve unless the diaphragm is damaged.

Plenty of you tubes how to clean and then test them with a $10 hand held vacuum pump.

So that's my 2 cents.

You seem to have ruled out a fuel delivery issue by spraying starter fluid into the intake and still getting a "No Start"

To completely rule a fuel delivery issue out you'll want to test the connection to one of the fuel injectors for power and ground signal from the computer with a multimeter.

(Here's how: https://itstillruns.com/test-injector-wiring-7995072.html )

Next you'll want to get an ignition spark tester and check for spark while cranking.

If you have power/ground to your injectors and spark to all of your plugs,

I'm giving the "fish eye" squarely at a stuck open EGR valve.

You won't find any "shorts to ground" by disconnecting the battery ground cable. You're removing the ground.

It sounds like by removing the negative battery cable from the post and then connecting your test light to the negative battery post and "picking" at a chassis ground point you were running the negative chassis ground from the negative battery post through your test light. You turned on the headlights, increased voltage draw through the tool,  and the test light got brighter.

If you're concerned that the DPFE sensor's melting damaged the wiring then, like the moderator said, check your fuse boxes for any blown fuses.

If you don't find any, to further allay your concerns, maybe check the wire connection at the DPFE sensor.

You Can Not do this with a test light. You have to USE a multimeter. Testing the computer's 5 volt reference signal and ground with a test light could damage the computer.

Get yourself a Tpin to "back probe" the DPFE sensor connector while it's plugged into the DPFE sensor.

On your DPFE connector, the bottom wire is the 5 volt reference signal from the computer.

Stick the Tpin into this wire's connection on the socket.

So set your multimeter to: DC Volts. Attach the red probe of your multimeter to the Tpin that you're back probing the bottom wire on the DPFE connector, and the multimeter's black probe to the battery negative post.

Turn the Key to ON and you should read about 5 volts+- here from the computer.

The middle wire is ground for the 5 volt reference signal. It's also provided by the computer.

So move your "back probe Tpin" to that wire.

(Still on DC volts) Attach the black multimeter probe to the Tpin and the red multimeter probe to the battery positive post.

Turn the Key to ON and you should read 12 volts.

The top wire is the signal wire from the DPFE sensor. Move your Tpin back probe there.

You can test it by removing those 2 small hoses from the DPFE sensor and attaching a length of hose of the same diameter to the forward hose attachment of the sensor.

(Still on DC volts) Attach the red probe of your multimeter to the Tpin and the black probe of your multimeter to the battery negative post.

Key On, Slowly suck (giggity) (apply suction) to that length of hose you attached to the front hose attachment of the DPFE sensor.

The voltage on your multimeter should move from around 0.9+- volts to around 5 volts+- as suction increases.

If all that checks out then it's unlikely that any electric damage occurred when your DPFE sensor melted.

But again, your DPFE sensor isn't supposed to melt. You should address that cause,.

lights up even more when I turn the headlight switch

it's supposed to.

Short circuits are extinguished by fuses. Do you have any blown fuses?