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.