Do switching type AC adaptors limit the current by themselves?
Without going into too much detail on the hows and whys and variations, for all intents and purposes the answer is yes, power supplies will only supply up to what they are rated to supply.  

Basically, if you have a 12V 5A supply, you can't try and draw more than 5A without the output voltage sagging, or some current limiting kicking in, fuse blowing, or smoke appearing. Usually with a well designed supply it will be current limiting or thermal fusing.   

So if you have a computer that has a 120W power unit (I take it this is a mini ATX type thing) then you need at least a 120W adaptor. If the unit runs from 12V, then you need 120W / 12V = 10A minimum. If you use the 60W adaptor with a 120W ATX, you effectively have a (slightly less than due to inefficiency) 60W ATX, as you can only get out what you put in.
So you need a new switching adaptor.