I would do a real close check on the battery cables, making sure that they are super clean and tight, not only the clamps on the battery, but also the clamps to the cables. Then I would clean the other ends of the cables making sure they are tight as well. If you are doubtful of the solenoid I would suggest replacing that as well and then check out the operation of the start switch by doing a by-pass.
As an aside, the engine will not suck fuel out of the tank, the tank is too low for that to happen, so your fuel pump must have been working to some extent, or you might have plugged filters.