Learn English – “falling due” vs “due”

The idiom is actually “to make do”, and it means to work with what you have, to continue somehow despite an impediment or non-ideal circumstance.

It uses do in the sense of “suffice”, as in “That’ll do”.

From The Encyclopedia of Word and Phrase Origins, by Robert Hendrickson:

The original version of this expression, 'on the water wagon' or 'water cart,' which isn't heard anymore, best explains the phrase. During the late 19th century, water carts drawn by horses wet down dusty roads in the summer. At the height of the Prohibition crusade in the 1890s men who vowed to stop drinking would say that they were thirsty indeed but would rather climb aboard the water cart to get a drink than break their pledges. From this sentiment came the expression 'I'm on the water cart,' I'm trying to stop drinking, which is first recorded in, of all places, Alice Caldwell Rice's Mrs. Wiggs of the Caggage Patch [1901], where the consumptive Mr. Dick says it to old Mrs. Wiggs. The more alliterative 'wagon' soon replaced cart in the expression and it was eventually shortened to 'on the wagon.' 'Fall off the (water) wagon' made its entry into the language almost immediately after its abstinent sister.