Microphone issue

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.

Br, you rock! Thank you so much for these great instructions. I'll test as you suggest tonight.

gswitz
https://www.radioshack.com/products/radioshack-15-range-digital-multimeter
 
I have this meter. I have a fresh battery in the Ohm Meter.
  
I broke out my SM57 for the purpose of testing and by touching the nodes to two of the prongs that go into the mic cable on the SM57 I can get the meter to move. I was also able to open the mic and get the meter to move with the prong tips on the solder nodes.
 
I am not able to get the meter to move on the ATM 25 either at the prongs or at the solder nodes near the capsule.

I can get a connection between the left pin and the white solder point and the center pin and the black solder point.
 
So, given the above, this means a bad coil.

It does sound that way, unfortunately. You might contact AT and see if they'll repair it for you.

Often it is not worth repairing either.  I have got an SM58 that has a distorting capsule.  They want around $120 for the replacement capsule but the whole mic we can get for around $150 here so not really worth the trouble.  A new one means a nice new fresh looking body, windshield etc.. New clip etc...