Electrical – DC Motor consuming too much current

To measure current with a basic multimeter, you have to break the circuit and insert the ammeter in series. That, is you must a series connection the 5V -> circuit board -> ammeter -> ground. If you used the ammeter in parallel, which your description suggests, then low resistance ammeter would draw as much current as the power supply can provide. This certainly could overheat the probe, although ammeters usually contain a fuse that should prevent this.

It's also possible you accidentally shorted power to ground with the probe somehow, or if a second power supply is involved (quite likely for microprocessors), that you created a connection between the power supplies.

By contrast, voltage is measured in parallel, and voltmeters are high resistance so they do not (usually) interfere with a circuit's operation.

It is often easier to place a "shunt" resistor in series with your load and measure the voltage as an indirect way of measuring current, because you do with this without breaking a circuit on a PCB.

What's important is to not overheat the motor or any of its parts (windings, brushes, bearings, etc.)

The heat in the motor comes from current-squared times resistance.

If the motor is rated for 1.95 amps max, then you have three options to avoid overheating it:

1. Make sure the load is never so high that the motor stalls or overloads.
2. Make sure that the voltage is so low that the current through the windings will never be higher than rated.
3. Use a current controller to drive the motor that can limit the current at the given maximum. (Typically, average current is limited through PWM.)

The reason your "2A power supply" didn't do the limiting is that it wasn't built with "continuous current limiting" as a feature. There exists power supplies that have this feature, but they are generally more expensive as it's usually harder to build that feature than either an unlimited power supply (that you can destroy by overloading it) or a intermittent-limiting power supply (that turns off entirely on overload or overheat.)

The amp rating for a power supply is generally how many amps it can safely deliver -- NOT a limit or exact number, like the voltage rating. The reason is that you don't "push" amps into a load; the load "draws" amps based on the voltage you supply and its internal construction (resistance, impedance.)

My recommendation for you is to get a 12V/2A power supply and see how much current is drawn if you stall the motor. If < 2A, great! If > 2A, reduce voltage even more, until the current drawn when stalled is acceptable.

If that's not good enough, then get a power supply that lets you establish a current limit, where the response of that power supply is to reduce voltage until current is under that limit. You can buy these as "components" from places like Jameco or Digi-Key, or you can buy a cheap 18V/3A benctop power supply from Amazon (which will also come with handy digital read-outs.)