[COLOR="Blue"]I have used two bottles of Draino.[/COLOR]
If you put this is the dishwasher this was a very bad idea.
It could eat some of the parts also not very good for you if Draino residue ends up on your dishes.

Is it possible I need to replace the pump?
I doubt it.

Would it be draining at the end of a cycle or manually if the motor was bad?
No.
There is a proviso here because it depends on how long the drain cycle is.

How do you know that it is not draining during a cycle?

[COLOR="Blue"]We have an excessive amount of standing water on the bottom of the dishwasher (not immediately after a cycle however)[/COLOR]
Could you clarify what the above means.

The way I interpret it is that immediately after a cycle there is no water in the tub but later on there is.
If this is correct then it is probably not a drain problem but could be a water valve leaking problem or water from the house plumbing draining back into the unit which can usually be fixed by re-positioning the drain hose