How can I prevent a 556 timer IC from becoming very hot?
According to the datasheet for the NE556, the no load supply current should be a maximum of 30mA at 15V supply. Since you are reading 125mA and the unit is getting hot suggests something is wrong somewhere.  

I can't see any obvious issue with your schematic at a glance that would cause this. So a few things to check:


Check that there are no shorts/low resistances anywhere (particularly from each output   to ground)    
Check polarised capacitors are the right way round.
Check resistor values are correct (in particular R4)  
Try swapping the IC in case it's faulty
Make sure only the solenoid is pulling the other 1.75A when activated (e.g. put multimeter in series with it only)
Disconnect solenoid, test, then disconnect first timer from second, test, etc, until the current drops to a reasonable value.  


If you can't find anything, trying one of the other circuits suggested might be worth a go.