The solenoid cannot cause power drain, unless the igntion switch is held in the start position, as this is the only time it uses power.
A duff alternator can flatten the battery, if the diode pack(regulator rectifier) is iffy, it will alow current flow from the battery to the alternator.
Charge the battery, disconnect the main charge wire output connection from the alternator, normally held down with 13mm nut etc, and see what it is like then.
Even if the charge warning light is not connected it will still start to charge when revved enough, as there will be enough residual current produced to start the charge process.
If the battery was discharging that quick the leads would be getting very hot, are they ?
tabetha