All, I replaced my doorbell transformer and got 22V (which was odd since it is supposedly a 16V transformer) with no wires connected to the transformer. As soon as I connect the wires to the transfomer that run to the chime the voltage drops to 2V or so. What would that indicate?
Assuming you have a full mains voltage going into the transformer, then there is a short-circuit in the wiring (or, much less likely, the transformer is vastly underrated for the current drawn).
"Bell wire" has thin insulation, compared to mains wiring. A staple may be pinching the wires, or wires could have rubbed together, tearing the insulation. Sometimes mice or squirrels can cause this. Such a short could have burned out the old transformer.
Another possibility is a short in the chime, which then caused the contacts of the doorbell switch to fuse together.
Also, inspect your connections to the transformer. If there are multiple wires, e.g. from both a front and a back door switch, they could be misconnected.
If you can, check the resistance between the bell wires going to the transformer secondary. The resistance should be greater than 16 divided by the current rating of the transformer, or (16 x 16) divided by the VA rating. For example, if the transformer is rated 10 VA, then circuit resistance should be greater than 256/10, or ~26 ohms. (Not to worry if it's just a bit lower, though: you're measuring DC resistance, while AC reactance across a coil is higher.)
Don't be concerned about the 22 VAC open circuit output because the nominal voltage, 16 VAC, is at the rated load.
BTW, my compliments on checking the voltage on installation! Had you not done so, the transformer would have burned out again, or there is a small possibility of fire if a "short-circuit" is a few ohms and is by something flammable.
1: Your Transformer is rated at 16VAC for what input Voltage ? It is Normal to have a higher voltage than rated output if the Xformer is rated 110V input and your home has 130VAC. Perhaps the xformer is really an 18VAC output and you have 130VAC feeding it?
2: Connecting the xformer to the chime and you get an immediate voltage drop. Means you have a load on that transformer, you should not have any load until your door bell switch makes the connection. Your door bell switch might be stuck in the always made position (shorted On), you might hear a small chime when you first connect the transformer wire - but that is all you will hear maybe a quick low bing at first.
Disconnect your door bell switch - check the power from your transformer with it disconnected and then if your power is ok - you know the switch is most likely the culprit. I would verify that by ohming the switch - and also tapping my two wires (at the switch) together (making like a momentary switch) to see if I get chimes.
3: Other than that you have a short - that is PRIOR to the switch.
4: 3rd possibility is both switch and chime are bad but I highly doubt that scenario - I am just listing it as a possibility because it is possible however rare.