The dynamic mic is essentially the same thing as a Left or Right headphone. It has a similar thin diaphragm inside, and tiny wires to a connector. The rest is mechanical science.
Here's what I'd do first:
1. Use an Ohm meter to test both sides of the coil. That means one meter lead on each solder joint on each side of the mic capsule. You should see the meter show a relatively low reading (64-ohms or below, in most cases). You're not worried as much about exact measurements. If the meter shows no reading, or ".OL", then you have a bad coil, and the mic could be beyond repair. Check with AT to see if they can sell you a new capsule.
2. If the coil is good, check the connection between those same solder positions and the XLR pins. One should show about zero (0) ohms measurement, and one may show that reading in Step 1. You can sometimes use the continuity "beep" function if you can't watch the meter display. Only
one XLR pin to one of the two main solder connections should give you a "good" or zero (0) Ohm reading or a beep. HOWEVER, if you do, check the ohms. It should not be exactly zero ohms, but maybe 10 Ohms or higher if the coil is good. Recall in step 1 that a good coil will read a certain Ohm measurement, but not exactly zero. That would be closely the same here.
3. Ensure none of the three XLR pins are shorted together (as in you should not see 0 ohms between any two XLR pins). If you do, there's maybe a stray very tiny wire in the mic somewhere frayed and touching the mic casing.
DO NOT attempt to connect a 9v battery up to the mic to test it like a speaker, if you're familiar with that.
For that mic, pretty much the only things that typically goes wrong is, the coil solder for the diaphragm fails, one of the wires breaks or corrodes/oxidizes beyond serviceability, or a tiny wire is frayed, touching the microphone casing. Trying to resolder is tricky if you don't use the right temperature soldering iron, as you can burn up the coil (if it isn't broken already). To access the white goo, which is normally never a place of failure in a dynamic mic, you can use a heat gun and careful digging with a small screwdriver.