The top solenoid appears to be the isolator solenoid, or at least the emergency start solenoid. Pressing the dash switch connects the coil to the house battery positive and so connects the two batteries together. If it's also the isolator solenoid (which is often but not always the case), I suspect the dash switch would also have a connection to a switched circuit from the chassis that's on when the ignition is on, connected to the normally closed contact.
The bottom solenoid may be the starter solenoid. Where does the other big wire go from it?
I think you probably have a bad wire or bad connection somewhere, rather than a bad solenoid, though the latter is possible. Check the battery and ground wires carefully at both of their ends. I would almost bet money that one of them looks OK but is actually falling apart, at least internally.
