Supplying power to a 12V motor with 5V and 2A adapter
Motors want lots of current to start, they want lots, given the opportunity they will take lots, and this will cause your 2A powersuply to quit.


  I was wondering if I used a 5V, 2A power adapter (that I already have), will it only supply 2A even if the motor tries to draw more than that and will this damage the adapter by any chance?


Typically they go into over-current shoutdown if over-loaded.

But motors don't need lots of current to start they only want it, if you use a DC-DC converter that has a current limit you can keep the power supply happy and get the motor spinning.