Lincoln is using a metaphor to describe the current situation. If a tub has no bottom, then when you turn on the water. The water will run and damage the surrounding area. There isn't a literal bottom missing from a tub that is making Lincoln discouraged. He knows the situation he is in is dire. Hope this helps.

It means the situation has become extremely bad, often hopeless.

You have given a lot of context, but not enough to allow me to figure out what this metaphor means. One page might not be enough. I might need to read two or three pages before this page to give you a good answer. One thing that is clear is that Lincoln feels that he has run out of something. However, what that something is is not clear. He could be referring to a depletion of financial resources. He could be referring to a depletion of military supplies. He could be referring to a lack of soldiers.

if it's the old language, where the tub = the barrel, it may mean that the barrel's bottom was taken out and thus the liquid inside it all over the floor = risky or dangerous situation